Stochastic modelling of bacterial lag phase

Phenotypic-dependent variability and the emergence of tolerance in bacterial populations

Ecological and evolutionary dynamics have been historically regarded as unfolding at broadly separated timescales. However, these two types of processes are nowadays well-documented to intersperse much more tightly than traditionally assumed, especially in communities of microorganisms. Advancing the development of mathematical and computational approaches to shed novel light onto eco-evolutionary problems is a challenge of utmost relevance. With this motivation in mind, here we scrutinize recent experimental results showing evidence of rapid evolution of tolerance by lag in bacterial populations that are periodically exposed to antibiotic stress in laboratory conditions. In particular, the distribution of single-cell lag times-i.e., the times that individual bacteria from the community remain in a dormant state to cope with stress-evolves its average value to approximately fit the antibiotic-exposure time. Moreover, the distribution develops right-skewed heavy tails, revealing the presence of individuals with anomalously large lag times. Here, we develop a parsimonious individual-based model mimicking the actual demographic processes of the experimental setup. Individuals are characterized by a single phenotypic trait: their intrinsic lag time, which is transmitted with variation to the progeny. The model-in a version in which the amplitude of phenotypic variations grows with the parent's lag time-is able to reproduce quite well the key empirical observations. Furthermore, we develop a general mathematical framework allowing us to describe with good accuracy the properties of the stochastic model by means of a macroscopic equation, which generalizes the Crow-Kimura equation in population genetics. Even if the model does not account for all the biological mechanisms (e.g., genetic changes) in a detailed way-i.e., it is a phenomenological one-it sheds light onto the eco-evolutionary dynamics of the problem and can be helpful to design strategies to hinder the emergence of tolerance in bacterial communities. From a broader perspective, this work represents a benchmark for the mathematical framework designed to tackle much more general eco-evolutionary problems, thus paving the road to further research avenues.

Application of growth rate from kinetic model to calculate stochastic growth of a bacteria population at low contamination level

Traditional predictive microbiology is not suited for cell growth predictions for low-level contamination, where individual cell heterogeneity becomes apparent. Accordingly, we simulated a stochastic birth process of bacteria population using kinetic parameters. We predicted the variation in behavior of Salmonella enterica serovar Typhimurium cells at low inoculum density. The modeled cells were grown in tryptic soy broth at 25 °C. Kinetic growth parameters were first determined empirically for an initial cell number of 10⁴ cells. Monte Carlo simulation based on the growth kinetics and Poisson distribution for different initial cell numbers predicted the results of 50 replicate growth experiments with the initial cell number of 1, 10, and 64 cells. Indeed, measured behavior of 85% cells fell within the 95% prediction area of the simulation. The calculations link the kinetic and stochastic birth process with Poisson distribution. The developed model can be used to calculate the probability distribution of population size for exposure assessment and for the evaluation of a probability that a pathogen would exceed critical contamination level during food storage.

On habit and habitat

Scientific theories based on mathematical models are frequently used in sciences to reveal natural behaviour of systems and eventually to be able in predicting such behaviour once the system's parameters and relevant conditions are known and can be specified. The integration of accumulated theoretical as well as experimental knowledge allows us to present such a unifying theory underlying the equivalence between habit and habitat in population growth. While the focus of the initial development was derived from microorganisms, the theory is extended to other population types too. The biological interpretation of ‘inertia’ or ‘habit’-based processes is provided as a consequence of this theory, and its relationship to the population ‘resource utilization’ available in the ‘habitat’ is derived. This paper focuses on the link between the ‘resource utilization’, which is related to the ‘habitat’, and ‘biological inertia’, which is related to population ‘habit’. This link extends the context of population growth and predictive modelling of microorganisms.

Potential Cultivation of Lactobacillus pentosus from Human Breastmilk with Rapid Monitoring through the Spectrophotometer Method

The present study focused on the development of a new method to determine the lag phase of Lactobacillus in breast milk which was attained during the 1st, 3rd, and 6th month (M1, M3, and M6). The colonies’ phylogenetic analysis, derived from the 16S rRNA gene sequences, was evaluated with genus Lactobacillus pentosus and achieved a similarity value of 99%. Raman spectroscopy in optical densities of 600 nm (OD600) were used for six consecutive days to observe the changes of the cell growth rate. The values of OD600 were well fitted with the regression model. From this work, M1 was found to be the longest lag phase in 18 h, and it was 17% to 27% longer compared to M3 and M6, respectively. However, the samples of M3 and M6 showed the shortest duration in reaching 0.5 of OD600 nm (16 h) which was enhanced by 80% and 96% compared to M1, respectively. These studies will be of significance when applied in determining the bacteria growth curve and in assessing the growth behavior for the strain in human breast milk.

Wide lag time distributions break a trade-off between reproduction and survival in bacteria

Many microorganisms face a fundamental trade-off between reproduction and survival: Rapid growth boosts population size but makes microorganisms sensitive to external stressors. Here, we show that starved bacteria encountering new resources can break this trade-off by evolving phenotypic heterogeneity in lag time. We quantify the distribution of single-cell lag times of populations of starved Escherichia coli and show that population growth after starvation is primarily determined by the cells with shortest lag due to the exponential nature of bacterial population dynamics. As a consequence, cells with long lag times have no substantial effect on population growth resumption. However, we observe that these cells provide tolerance to stressors such as antibiotics. This allows an isogenic population to break the trade-off between reproduction and survival. We support this argument with an evolutionary model which shows that bacteria evolve wide lag time distributions when both rapid growth resumption and survival under stressful conditions are under selection. Our results can explain the prevalence of antibiotic tolerance by lag and demonstrate that the benefits of phenotypic heterogeneity in fluctuating environments are particularly high when minorities with extreme phenotypes dominate population dynamics.

Lag Phase Is a Dynamic, Organized, Adaptive, and Evolvable Period That Prepares Bacteria for Cell Division

Lag is a temporary period of nonreplication seen in bacteria that are introduced to new media. Despite latency being described by Müller in 1895, only recently have we gained insights into the cellular processes characterizing lag phase. This review covers literature to date on the transcriptomic, proteomic, metabolomic, physiological, biochemical, and evolutionary features of prokaryotic lag. Though lag is commonly described as a preparative phase that allows bacteria to harvest nutrients and adapt to new environments, the implications of recent studies indicate that a refinement of this view is well deserved. As shown, lag is a dynamic, organized, adaptive, and evolvable process that protects bacteria from threats, promotes reproductive fitness, and is broadly relevant to the study of bacterial evolution, host-pathogen interactions, antibiotic tolerance, environmental biology, molecular microbiology, and food safety.

Leucine-Based Polymer Architecture-Induced Antimicrobial Properties and Bacterial Cell Morphology Switching

To evaluate the comparative antibacterial activity of leucine-based cationic polymers having linear, hyperbranched, and star architectures containing both hydrophilic and hydrophobic segments against Gram-negative bacterium, Escherichia coli (E. coli), herein we performed zone of inhibition study, minimum inhibitory concentration (MIC) calculation, and bacterial growth experiment. The highest antibacterial activity in terms of the MIC value was found in hyperbranched and star architectures because of the greater extent of cationic and hydrophobic functionality, enhancing cell wall penetration ability compared to that of the linear polymer. The absence of the bacterial regrowth stage in the growth curve exhibited the highest bactericidal capacity of star polymers, when untreated cells (control) already reached to the stationary phase, whereas the bacterial regrowth stage with a delayed lag phase was critically observed for linear and hyperbranched architectures displaying lower bactericidal efficacy. Coagulation of E. coli cells, switching of cell morphology from rod to sphere, and lengthening due to stacking in an antimicrobial polymer-treated environment at the bacterial regrowth stage in liquid media were visualized critically by field emission scanning electron microscopy and confocal fluorescence microscopy instruments in the presence of 4',6-diamidino-2-phenylindole stain.

Effect of storage temperature on the lag time of Geobacillus stearothermophilus individual spores

The lag times (λ) of Geobacillus stearothermophilus single spores were studied at different storage temperatures ranging from 45 to 59 °C using the Bioscreen C method. A significant variability of λ was observed among individual spores at all temperatures tested. The storage temperature affected both the position and the spread of the λ distributions. The minimum mean value of λ (i.e. 10.87 h) was observed at 55 °C, while moving away from this temperature resulted in an increase for both the mean and standard deviation of λ. A Cardinal Model with Inflection (CMI) was fitted to the reverse mean λ, and the estimated values for the cardinal parameters T_min, T_max, T_opt and the optimum mean λ of G. stearothermophilus were found to be 38.1, 64.2, 53.6 °C and 10.3 h, respectively. To interpret the observations, a probabilistic growth model for G. stearothermophilus individual spores, taking into account λ variability, was developed. The model describes the growth of a population, initially consisting of N₀ spores, over time as the sum of cells in each of the N₀ imminent subpopulations originating from a single spore. Growth simulations for different initial contamination levels showed that for low N₀ the number of cells in the population at any time is highly variable. An increase in N₀ to levels exceeding 100 spores results in a significant decrease of the above variability and a shorter λ of the population. Considering that the number of G. stearothermophilus surviving spores in the final product is usually very low, the data provided in this work can be used to evaluate the probability distribution of the time-to-spoilage and enable decision-making based on the "acceptable level of risk".

The bio-physics of condensation of divalent cations into the bacterial wall has implications for growth of Gram-positive bacteria

BACKGROUND

The anionic-polyelectrolyte nature of the wall of Gram-positive bacteria has long been suspected to be involved in homeostasis of essential cations and bacterial growth. A better understanding of the coupling between the biophysics and the biology of the wall is essential to understand some key features at play in ion-homeostasis in this living system.

METHODS

We consider the wall as a polyelectrolyte gel and balance the long-range electrostatic repulsion within this structure against the penalty entropy required to condense cations around wall polyelectrolytes. The resulting equations define how cations interact physically with the wall and the characteristic time required for a cation to leave the wall and enter into the bacterium to enable its usage for bacterial metabolism and growth.

RESULTS

The model was challenged against experimental data regarding growth of Gram-positive bacteria in the presence of varying concentration of divalent ions. The model explains qualitatively and quantitatively how divalent cations interact with the wall as well as how the biophysical properties of the wall impact on bacterial growth (in particular the initiation of bacterial growth).

CONCLUSION

The interplay between polymer biophysics and the biology of Gram positive bacteria is defined for the first time as a new set of variables that contribute to the kinetics of bacterial growth.

GENERAL SIGNIFICANCE

Providing an understanding of how bacteria capture essential metal cations in way that does not follow usual binding laws has implications when considering the control of such organisms and their ability to survive and grow in extreme environments.

Modeling the Growth of Epiphytic Bacteria on Kale Treated by Thermosonication Combined with Slightly Acidic Electrolyzed Water and Stored under Dynamic Temperature Conditions

The growth of epiphytic bacteria (aerobic mesophilic bacteria or Pseudomonas spp.) on kale was modeled isothermally and validated under dynamic storage temperatures. Each bacterial count on kale stored at isothermal conditions (4 to 25 °C) was recorded. The results show that maximum growth rate (μmax ) of both epiphytic bacteria increased and lag time (λ) decreased with increasing temperature (P < 0.05). The maximum population density (Nmax ) of Pseudomonas spp. was significantly greater than that of aerobic mesophilic bacteria, particularly in treated samples and/or at 4 and 10 °C (P < 0.05). The relationship between μmax of both epiphytic bacteria and temperature was linear (R(2) > 0.97), whereas lower R(2) > 0.86 and R(2) > 0.87 was observed for the λ and Nmax , respectively. The overall predictions of both epiphytic bacterial growths under nonisothermal conditions with temperature abuse of 15 °C agreed with the observed data, whereas those with temperature abuse of 25 °C were greatly overestimated. The appropriate parameter q0 (physiological state of cells), therefore, was adjusted by a trial and error to fit the model. This study demonstrates that the developed model was able to predict accurately epiphytic bacterial growth on kale stored under nonisothermal conditions particularly those with low temperature abuse of 15 °C.

Quantifying the effect of water activity and storage temperature on single spore lag times of three moulds isolated from spoiled bakery products

The inhibitory effect of water activity (a_w) and storage temperature on single spore lag times of Aspergillus niger, Eurotium repens (Aspergillus pseudoglaucus) and Penicillium corylophilum strains isolated from spoiled bakery products, was quantified. A full factorial design was set up for each strain. Data were collected at levels of a_w varying from 0.80 to 0.98 and temperature from 15 to 35°C. Experiments were performed on malt agar, at pH5.5. When growth was observed, ca 20 individual growth kinetics per condition were recorded up to 35days. Radius of the colony vs time was then fitted with the Buchanan primary model. For each experimental condition, a lag time variability was observed, it was characterized by its mean, standard deviation (sd) and 5^th percentile, after a Normal distribution fit. As the environmental conditions became stressful (e.g. storage temperature and a_w lower), mean and sd of single spore lag time distribution increased, indicating longer lag times and higher variability. The relationship between mean and sd followed a monotonous but not linear pattern, identical whatever the species. Next, secondary models were deployed to estimate the cardinal values (minimal, optimal and maximal temperatures, minimal water activity where no growth is observed anymore) for the three species. That enabled to confirm the observation made based on raw data analysis: concerning the temperature effect, A. niger behaviour was significantly different from E. repens and P. corylophilum: T_opt of 37.4°C (standard deviation 1.4°C) instead of 27.1°C (1.4°C) and 25.2°C (1.2°C), respectively. Concerning the a_w effect, from the three mould species, E. repens was the species able to grow at the lowest a_w (aw_min estimated to 0.74 (0.02)). Finally, results obtained with single spores were compared to findings from a previous study carried out at the population level (Dagnas et al., 2014). For short lag times (≤5days), there was no difference between lag time of the population (ca 2000 spores inoculated in one spot) and mean (nor 5^th percentile) of single spore lag time distribution. In contrast, when lag time was longer, i.e. under more stressful conditions, there was a discrepancy between individual and population lag times (population lag times shorter than 5^th percentiles of single spore lag time distribution), confirming a stochastic process. Finally, the temperature cardinal values estimated with single spores were found to be similar to those obtained at the population level, whatever the species. All these findings will be used to describe better mould spore lag time variability and then to predict more accurately bakery product shelf-life.

Variability in Cell Response of Cronobacter sakazakii after Mild-Heat Treatments and Its Impact on Food Safety

Cronobacter spp. have been responsible for severe infections in infants associated with consumption of powdered infant formula and follow-up formulae. Despite several risk assessments described in published studies, few approaches have considered the tremendous variability in cell response that small micropopulations or single cells can have in infant formula during storage, preparation or post process/preparation before the feeding of infants. Stochastic approaches can better describe microbial single cell response than deterministic models as we prove in this study. A large variability of lag phase was observed in single cell and micropopulations of ≤50 cells. This variability increased as the heat shock increased and growth temperature decreased. Obviously, variability of growth of individual Cronobacter sakazakii cell is affected by inoculum size, growth temperature and the probability of cells able to grow at the conditions imposed by the experimental conditions should be taken into account, especially when errors in bottle-preparation practices, such as improper holding temperatures, or manipulation, may lead to growth of the pathogen to a critical cell level. The mean probability of illness from initial inoculum size of 1 cell was below 0.2 in all the cases and for inoculum size of 50 cells the mean probability of illness, in most of the cases, was above 0.7.

Towards lag phase of microbial populations at growth-limiting conditions: The role of the variability in the growth limits of individual cells

The water activity (aw) growth limits of unheated and heat stressed Listeria monocytogenes individual cells were studied. The aw limits varied from 0.940 to 0.997 and 0.951 to 0.997 for unheated and heat stressed cells, respectively. Due to the above variability a decrease in aw results in the presence of a non-growing fraction in the population leading to an additional pseudo-lag in population growth. In this case the total apparent lag of the population is the sum of the physiological lag of the growing cells (time required to adjust to the new environment) and the pseudo-lag. To investigate the effect of aw on the above lag components, the growth kinetics of L. monocytogenes on tryptone soy agar with aw adjusted to values ranging from 0.997 to 0.940 was monitored. The model of B&R was fitted to the data for the estimation of the apparent lag. In order to estimate the physiological lag of the growing fraction of the inoculum, the model was refitted to the growth data using as initial population level the number of cells that were able to grow (estimated from the number of colonies formed on the agar at the end of storage) and excluding the rest data during the lag. The results showed that for the unheated cells the apparent lag was almost identical to the physiological lag for aw values ranging from 0.997 to 0.970, as the majority of the cells in the initial population was able to grow in these conditions. As the aw decreased from 0.970 to 0.940 however, the number of cells in the population which were able to grow, decreased resulting to an increase in the pseudo-lag. The maximum value of pseudo-lag was 13.1h and it was observed at aw=0.940 where 10% of the total inoculated cells were able to grow. For heat stressed populations a pseudo-lag started to increase at higher aw conditions (0.982) compared to unheated cells. In contrast to the apparent lag, a linear relation between physiological lag and aw was observed for both unheated and heat stressed cells.

Bridging the divide: a model-data approach to Polar and Alpine microbiology

Advances in microbial ecology in the cryosphere continue to be driven by empirical approaches including field sampling and laboratory-based analyses. Although mathematical models are commonly used to investigate the physical dynamics of Polar and Alpine regions, they are rarely applied in microbial studies. Yet integrating modelling approaches with ongoing observational and laboratory-based work is ideally suited to Polar and Alpine microbial ecosystems given their harsh environmental and biogeochemical characteristics, simple trophic structures, distinct seasonality, often difficult accessibility, geographical expansiveness and susceptibility to accelerated climate changes. In this opinion paper, we explain how mathematical modelling ideally complements field and laboratory-based analyses. We thus argue that mathematical modelling is a powerful tool for the investigation of these extreme environments and that fully integrated, interdisciplinary model-data approaches could help the Polar and Alpine microbiology community address some of the great research challenges of the 21st century (e.g. assessing global significance and response to climate change). However, a better integration of field and laboratory work with model design and calibration/validation, as well as a stronger focus on quantitative information is required to advance models that can be used to make predictions and upscale processes and fluxes beyond what can be captured by observations alone.

Colonial vs. planktonic type of growth: mathematical modeling of microbial dynamics on surfaces and in liquid, semi-liquid and solid foods

Predictive models are mathematical expressions that describe the growth, survival, inactivation, or biochemical processes of foodborne bacteria. During processing of contaminated raw materials and food preparation, bacteria are entrapped into the food residues, potentially transferred to the equipment surfaces (abiotic or inert surfaces) or cross-contaminate other foods (biotic surfaces). Growth of bacterial cells can either occur planktonically in liquid or immobilized as colonies. Colonies are on the surface or confined in the interior (submerged colonies) of structured foods. For low initial levels of bacterial population leading to large colonies, the immobilized growth differs from planktonic growth due to physical constrains and to diffusion limitations within the structured foods. Indeed, cells in colonies experience substrate starvation and/or stresses from the accumulation of toxic metabolites such as lactic acid. Furthermore, the micro-architecture of foods also influences the rate and extent of growth. The micro-architecture is determined by (i) the non-aqueous phase with the distribution and size of oil particles and the pore size of the network when proteins or gelling agent are solidified, and by (ii) the available aqueous phase within which bacteria may swarm or swim. As a consequence, the micro-environment of bacterial cells when they grow in colonies might greatly differs from that when they grow planktonically. The broth-based data used for modeling (lag time and generation time, the growth rate, and population level) are poorly transferable to solid foods. It may lead to an over-estimation or under-estimation of the predicted population compared to the observed population in food. If the growth prediction concerns pathogen bacteria, it is a major importance for the safety of foods to improve the knowledge on immobilized growth. In this review, the different types of models are presented taking into account the stochastic behavior of single cells in the growth of a bacterial population. Finally, the recent advances in the rules controlling different modes of growth, as well as the methodological approaches for monitoring and modeling such growth are detailed.

Modeling red cabbage seed extract effect on Penicillium corylophilum: Relationship between germination time, individual and population lag time

The inhibitory effect of a red cabbage seed extract on germination time, individual (single spore) and population lag time of Penicillium corylophilum was studied. First, to compare the biological variability of single spore germination and lag times under stressful conditions, data were collected at levels of red cabbage seed extract varying from 0 to 10 mg/g (150 spores observed in each trial of germination, ca 50 spores in each individual lag experiment). Experiments were performed on malt agar at 25 °C, pH 5.2, aw 0.99. The data, without any transformation, were statistically analyzed; several probability distribution functions were used to fit the cumulated germination times and the individual lag times of spores. In both cases, the best fit was obtained with the Normal distribution. In parallel, lag times at the population level (ca 2000 spores per trial) were collected for the same range of plant extract. Not surprisingly, the difference between individual and population lag times could be explained by a stochastic process. More interestingly, it was shown that under stressful conditions, the population lag time did not correspond to the time required for germination of 95% of spores, but to a much longer time. Finally, it was deduced from the statistical analysis, completed by microscopic observations, that the plant extract affected mainly the hyphal elongation (and then the lag time) and not the germination. Next, secondary models were developed to quantify the effect of red cabbage seed extract on the median of germination times, individual and population lag times. The Minimum Inhibitory Concentrations (MICs) were estimated. It was shown that the red cabbage seed extract MIC for P. corylophilum lag time did not depend on the inoculum load. Application of the secondary models allowed us to conclude that under the conditions of our experiment, the addition of 10 mg/g of red cabbage seed extract enabled extension of lag time to two weeks.

Investigation on the Factors Affecting Cronobacter sakazakii Contamination Levels in Reconstituted Powdered Infant Formula

INTRODUCTION

Certain strains of Cronobacter sakazakii can cause serious invasive infections in children, mainly those <2 months old and fed with powdered infant formula (PIF). The infectious dose of C. sakazakii is unknown but evidence suggests that it is approximately 1000 colony forming units (CFU). PIF is currently considered safe if its end-product C. sakazakii level is <1 CFU/g. In this study, we determined the lag time, generation time (GT), and growth rate of five pooled C. sakazakii isolates to evaluate the factors affecting contamination levels in reconstituted PIF.

METHODS

1.71 log CFU/ml of C. sakazakii were inoculated into 100 and 3000 ml of reconstituted PIF and incubated at 22 and 35°C. Growth was evaluated over a 24-h period. ComBase was used for modeling.

RESULTS

In 3000 ml, the growth rate was 0.45 ± 0.02 log CFU/h with a lag phase of 3 ± 0.05 h and GT of 0.67 h at 22°C, while the growth rate was 0.73 ± 0.01 log CFU/h with a lag phase of 0.45 ± 0.03 h and GT of 0.41 h at 35° C.

CONCLUSION

Cronobacter sakazakii grows rapidly in reconstituted PIF, especially at 35° C.

Assessing the survival of Listeria monocytogenes in a domestic freezer by analyzing subsequent growth at 30°C using a novel reference method

Listeria monocytogenes is a serious pathogen capable of extensive survival under frozen storage. Using optical density and multiple initial inocula in multiple identically prepared microtiter plates, the effect of storage time at -22°C on the subsequent growth at 30°C of the organism when defrosted was studied using a technique that compared the growth (through time to detection) of a test plate (previously frozen) with that of an identically prepared control plate, analyzed at the start of the experiment. Experiments were carried out using tryptic soy broth (TSB) or TSB supplemented with 3% salt. Plates were stored and frozen for up to 6 months (10 days, 20 days, 2 months, and 6 months). As storage time increased, there was only a small relative increase in the lag and the variance in the time to detection observed. When compared with storage in 3% salt TSB, which reduced the specific growth rate relative to growth in standard TSB, there were only marginally greater increases in lag and data variance. After 6 months storage in 3% salt TSB, there were some indications of inactivation (observed as small reductions of the initial optical density (equal to 1 × 10(9) CFU/ml) equivalent to a 50% inactivation. The method and the analyses suggest that this technique could allow easy examination of the effect of frozen storage on given cultures, with respect to the effects of pH, water activity, and also the effect of preservatives commonly used as extra hurdles in foods.

Modeling bacterial population growth from stochastic single-cell dynamics

A few bacterial cells may be sufficient to produce a food-borne illness outbreak, provided that they are capable of adapting and proliferating on a food matrix. This is why any quantitative health risk assessment policy must incorporate methods to accurately predict the growth of bacterial populations from a small number of pathogens. In this aim, mathematical models have become a powerful tool. Unfortunately, at low cell concentrations, standard deterministic models fail to predict the fate of the population, essentially because the heterogeneity between individuals becomes relevant. In this work, a stochastic differential equation (SDE) model is proposed to describe variability within single-cell growth and division and to simulate population growth from a given initial number of individuals. We provide evidence of the model ability to explain the observed distributions of times to division, including the lag time produced by the adaptation to the environment, by comparing model predictions with experiments from the literature for Escherichia coli, Listeria innocua, and Salmonella enterica. The model is shown to accurately predict experimental growth population dynamics for both small and large microbial populations. The use of stochastic models for the estimation of parameters to successfully fit experimental data is a particularly challenging problem. For instance, if Monte Carlo methods are employed to model the required distributions of times to division, the parameter estimation problem can become numerically intractable. We overcame this limitation by converting the stochastic description to a partial differential equation (backward Kolmogorov) instead, which relates to the distribution of division times. Contrary to previous stochastic formulations based on random parameters, the present model is capable of explaining the variability observed in populations that result from the growth of a small number of initial cells as well as the lack of it compared to populations initiated by a larger number of individuals, where the random effects become negligible.

Effect of inoculum size and water activity on the time to visible growth of Penicillium chrysogenum colony

In order to assess the effect of the inoculum size on the time to visible growth for Penicillium chrysogenum, the correlation described by González et al. (González, H.H.L., Resnik, S.L., Vaamonde, G., 1987. Influence of inoculum size on growth rate and lag phase of fungi isolate from Argentine corn. International Journal of Food Microbiology 4, 111-117) was compared to the model introduced by Gougouli et al. (Gougouli, M., Kalantzi, K., Beletsiotis, E., Koutsoumanis, K.P., 2011. Development and application of predictive models for fungal growth as tools to improve quality control in yogurt production. Food Microbiology 28, 1453-1462). Based on the regression coefficient, the latter model performed better than the former one to fit the data obtained for P. chrysogenum grown on Potato Dextrose Agar at 25 °C. Inoculum sizes in the range 10(1)-10(5) spores were tested at 0.930, 0.950, 0.970, and 0.995 aw. By extrapolation of the straight line, the model of Gougouli et al. (2011) provided accurate estimations of the time to visible growth for a single spore inoculum, tvg (N=1). In order to avoid experiments at reduced water activities, the influence of water activity on the model parameters, and on the ratio tvg (N=1) over the germination time was assessed.

Stochasticity in colonial growth dynamics of individual bacterial cells

Conventional bacterial growth studies rely on large bacterial populations without considering the individual cells. Individual cells, however, can exhibit marked behavioral heterogeneity. Here, we present experimental observations on the colonial growth of 220 individual cells of Salmonella enterica serotype Typhimurium using time-lapse microscopy videos. We found a highly heterogeneous behavior. Some cells did not grow, showing filamentation or lysis before division. Cells that were able to grow and form microcolonies showed highly diverse growth dynamics. The quality of the videos allowed for counting the cells over time and estimating the kinetic parameters lag time (λ) and maximum specific growth rate (μmax) for each microcolony originating from a single cell. To interpret the observations, the variability of the kinetic parameters was characterized using appropriate probability distributions and introduced to a stochastic model that allows for taking into account heterogeneity using Monte Carlo simulation. The model provides stochastic growth curves demonstrating that growth of single cells or small microbial populations is a pool of events each one of which has its own probability to occur. Simulations of the model illustrated how the apparent variability in population growth gradually decreases with increasing initial population size (N(0)). For bacterial populations with N(0) of >100 cells, the variability is almost eliminated and the system seems to behave deterministically, even though the underlying law is stochastic. We also used the model to demonstrate the effect of the presence and extent of a nongrowing population fraction on the stochastic growth of bacterial populations.

Stochastic modeling of Pseudomonas syringae growth in the phyllosphere

Pseudomonas syringae is a gram-negative bacterium which lives on leaf surfaces. Its growth has been described using epifluorescence microscopy and image analysis; it was found to be growing in aggregates of a wide range of sizes. We develop a stochastic model to describe aggregate distribution and determine the mechanisms generating experimental observations. We found that a logistic birth-death model with migration (time-homogeneous Markov process) provides the best description of the observed data. We discuss how to analyze the joint distribution of the numbers of aggregates of different sizes at a given time and explore how to account for new aggregates being created, that is, the joint distribution of the family size statistics conditional on the total number of aggregates. We compute the first two moments. Through simulations we examine how the model's parameters affect the aggregate size distribution and successfully explain the quantitative experimental data available. Aggregation formation is thought to be the first step towards pathogenic behavior of this bacterium; understanding aggregate size distribution would prove useful to understand the switch from epiphytic to pathogenic behavior.

Modeling and estimating bacterial lag phase

A branching process model of a bacterial population with initial lag phase is developed. Approximations are established in order to facilitate parameter estimation. The validity of approximations and estimation procedures is tested with simulated data.

Can telomere shortening explain sigmoidal growth curves?

A general branching process model is proposed to describe the shortening of telomeres in eukaryotic chromosomes. The model is flexible and incorporates many special cases to be found in the literature. In particular, we show how telomere shortening can give rise to sigmoidal growth curves, an idea first expressed by Portugal et al. [A computational model for telomere-dependent cell-replicative aging, BioSystems 91 (2008), pp. 262-267]. We also demonstrate how other types of growth curves arise if telomere shortening is mitigated by other cellular processes. We compare our results with published data sets from the biological literature.

Exploring the lag phase and growth initiation of a yeast culture by means of an individual-based model

The performance of fermentation processes is greatly influenced by the size and quality of inocula. The characterization of the replicative age is decided by the number of birth scars each yeast exhibits on its cellular membrane. Yeast ageing and inoculum size are factors that affect industrial fermentation, particularly those processes in which the yeast cells are reused such as the production of beer. This process reuses yeast cropped at the end of one fermentation in the following one, in a process called "serial repitching". The aim of this study was to explore the effects of inoculum size and ageing on the first stages of the dynamics of yeast population growth. However, only Individual-based Models (IbMs) allow the study of small, well-characterized, microbial inocula. We used INDISIM-YEAST, based on the generic IbM simulator INDISIM, to carry out these studies. Several simulations were performed to analyze the effect of the inoculum size and genealogical age of the cells that made it up on the lag phase, first division time and specific growth rate. The shortest lag phase and time to the first division were obtained with largest inocula and with the youngest inoculated parent cells.

Probabilistic model of microbial cell growth, division, and mortality

After a short time interval of length deltat during microbial growth, an individual cell can be found to be divided with probability Pd(t)deltat, dead with probability Pm(t)deltat, or alive but undivided with the probability 1-[Pd(t)+Pm(t)]deltat, where t is time, Pd(t) expresses the probability of division for an individual cell per unit of time, and Pm(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat's properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the population's size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a "shoulder," and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With Pd(t) and Pm(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.

Safety-based shelf life model for frankfurters based on time to detect Listeria monocytogenes with initial inoculum below detection limit

The growth of Listeria monocytogenes inoculated on frankfurters at four inoculum levels (0.1, 0.04, 0.01, and 0.007 CFU/g) was examined at 4, 8, and 12 degrees C until the time L. monocytogenes populations reached a detectable limit of at least 2 CFU/g. A scaled-down assumption was made to simulate a 25-g sample from a 100-lb batch size in a factory setting by using a 0.55-g sample from a 1,000-g batch size in a laboratory. Samples of 0.55 g were enriched in PDX-LIB selective medium, and presumptive results were confirmed on modified Oxford agar. Based on the time to detect (TTD) from each inoculum level and at each temperature, a shelf life model was constructed to predict the detection or risk levels reached by L. monocytogenes on frankfurters. The TTD increased with reductions in inoculum size and storage temperature. At 4 degrees C the TTDs (+/- standard error) observed were 42.0 +/- 1.0, 43.5 +/- 0.5, 50.7 +/- 1.5, and 55.0 +/- 3.0 days when the inoculum sizes were 0.1, 0.04, 0.01, and 0.007 CFU/g, respectively. From the same corresponding inoculum sizes, the TTDs at 8 degrees C were 4.5 +/- 0.5, 6.5 +/- 0.5, 7.0 +/- 1.0, and 8.5 +/- 0.5 days. Significant differences (P < 0.05) between TTDs were observed only when the inoculum sizes differed by at least 2 log. On a shelf life plot of 1n (TTD) versus temperature, the Q10 (increase in TTD for a 10 degrees C increase in temperature) values ranged from 24.5 to 44.7 and with no significant influence from the inoculum densities. When the observed TTDs were compared with the expected detection times based on the data obtained from a study with an inoculum size of 10 to 20 CFU/g, significant deviations were noted at lower inoculum levels. These results can be valuable in designing a safety-based shelf life model for frankfurters and in performing quantitative risk assessment of listeriosis at low and practical contamination levels.

Effect of capric, lauric and alpha-linolenic acids on the division time distributions of single cells of Staphylococcus aureus

The effect of non-inhibitory concentrations of capric, lauric and alpha-linolenic acids (C10:0, C12:0 and C18:3 respectively) on the division time distribution of single cells of Staphylococcus aureus was evaluated at pH 7 and pH 5. The effect of the initial cell concentration on the lag time of growing cell populations was also assessed. The statistical properties of the division times (defined as the time interval from birth to next binary fission for a single cell) were studied using the method of Elfwing et al. [Elfwing, A., Le Marc, Y., Baranyi, J., Ballagi, A., 2004. Observing the growth and division of large number of individual bacteria using image analysis. Applied and Environmental Microbiology 70, 675-678]. The division times were significantly longer in the presence of free fatty acids than in the control. Shorter division intervals were detected at pH 7 than at pH 5 in the control experiment and in the presence of C10:0. However, both C12:0 and C18:3 slowed down the growth, regardless of the pH. The observed division time distributions were used to simulate growth curves from different inoculum sizes using the stochastic birth process described by Pin and Baranyi [Pin, C., Baranyi, J., 2006. Kinetics of single cells: observation and modelling of a stochastic process. Applied and Environmental Microbiology 72, 2163-2169]. The output of the simulation results were compared with observed data. The lag times fitted to simulated growth curves were in good agreement with those fitted to growth curves measured by plate counts. The averaged out effect of the population masked the effect of the free fatty acids and pH on the division times of single cells.