Connection between stochastic and deterministic modelling of microbial growth

We present in this paper various links between individual and population cell growth. Deterministic models of the lag and subsequent growth of a bacterial population and their connection with stochastic models for the lag and subsequent generation times of individual cells are analysed. We derived the individual lag time distribution inherent in population growth models, which shows that the Baranyi model allows a wide range of shapes for individual lag time distribution. We demonstrate that individual cell lag time distributions cannot be retrieved from population growth data. We also present the results of our investigation on the effect of the mean and variance of the individual lag time and the initial cell number on the mean and variance of the population lag time. These relationships are analysed theoretically, and their consequence for predictive microbiology research is discussed.     

Parameter estimation for the distribution of single cell lag times

In Quantitative Microbial Risk Assessment, it is vital to understand how lag times of individual cells are distributed over a bacterial population. Such identified distributions can be used to predict the time by which, in a growth-supporting environment, a few pathogenic cells can multiply to a poisoning concentration level.We model the lag time of a single cell, inoculated into a new environment, by the delay of the growth function characterizing the generated subpopulation. We introduce an easy-to-implement procedure, based on the method of moments, to estimate the parameters of the distribution of single cell lag times. The advantage of the method is especially apparent for cases where the initial number of cells is small and random, and the culture is detectable only in the exponential growth phase.     

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.     

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.     

Studies on colony formation in vitro by mouse bone marrow cells. II. Action of colony stimulating factor

An analysis was made of some of the processes involved in the stimulation by colony stimulating factor (CSF) of cluster and colony formation by mouse bone marrow cells in agar cultures in vitro. Colony formation was shown to be related to the concentration and not the total amount of CSF. The concentration of CSF determined the rate of new cluster initiation in cultures and the rate of growth of individual clusters. Colony growth depleted the medium of CSF suggesting that colony cells may utilise CSF during proliferation. Bone marrow cells incubated in agar in the absence of CSF rapidly died or lost their capacity to proliferate and form clusters or colonies. CSF appears (a) to be necessary for survival of cluster-and colony-forming cells or for survival of their proliferative potential, (b) to shorten the lag period before individual cells commence proliferation and (c) to increase the growth rate of individual clusters and colonies.     

Effect of growth phase on survival of bromegrass suspension cells following cryopreservation and abiotic stresses

BACKGROUND AND AIMS: Cryopreservation is a practical method of preserving plant cell cultures and their genetic integrity. It has long been believed that cryopreservation of plant cell cultures is best performed with cells at the late lag or early exponential growth phase. At these stages the cells are small and non-vacuolated. This belief was based on studies using conventional slow prefreezing protocols and survival determined with fluorescein diacetate staining or 2,3,5-triphenyltetrazolium chloride assays. This classical issue was revisited here to determine the optimum growth phase for cryopreserving a bromegrass (Bromus inermis) suspension culture using more recently developed protocols and regrowth assays for determination of survival. METHODS: Cells at different growth phases were cryopreserved using three protocols: slow prefreezing, rapid prefreezing and vitrification. Stage-dependent trends in cell osmolarity, water content and tolerance to freezing, heat and salt stresses were also determined. In all cases survival was assayed by regrowth of cells following the treatments. KEY RESULTS: Slow prefreezing and rapid prefreezing protocols resulted in higher cell survival compared with the vitrification method. For all the protocols used, the best regrowth was obtained using cells in the late exponential or early stationary phase, whereas lowest survival was obtained for cells in the late lag or early exponential phase. Cells at the late exponential phase were characterized by high water content and high osmolarity and were most tolerant to freezing, heat and salt stresses, whereas cells at the early exponential phase, characterized by low water content and low osmolarity, were least tolerant. CONCLUSIONS: The results are contrary to the classical concept which utilizes cells in the late lag or early exponential growth phase for cryopreservation. The optimal growth phase for cryopreservation may depend upon the species or cell culture being cryopreserved and requires re-investigation for each cell culture. Stage-dependent survival following cryopreservation was proportionally correlated with the levels of abiotic stress tolerance in bromegrass cells.     

Assessing the Contamination Potential of Freshly Extracted Escherichia coli Biofilm Cells by Impedancemetry

Planktonic bacteria passing to a sessile state during the formation of a biofilm undergo many gene expression and phenotypic changes. These transformations require a significant time to establish. Inversely, cells extracted from a biofilm should also require a significant time before acquiring the same physiological characteristics as planktonic cells. Relatively few studies have addressed the kinetics of this inverse transformation process. We tested one aspect, namely, the contamination potential of freshly extracted Escherichia coli biofilm cells, precultured in a synthetic medium, in a rich liquid growth medium. We compared the time between inoculation and the beginning of the growth phase of freshly extracted biofilm cells, and suspended exponential and suspended stationary phase cells precultured in the same synthetic medium. Unexpectedly, the lag time for the extracted biofilm cells was the same as the lag time of the suspended exponential phase cells and significantly less than the lag time of the suspended stationary phase cells. The lag times were determined by an impedance technique. Cells extracted from biofilms, i.e., biofilms formed in canalizations and broken up by hydrodynamic forces, are an important source of contamination. Our work shows, in the case of E. coli, the high potential of freshly extracted biofilm cells to reinfect a new medium.     

Estimating Bacterial Growth Parameters by Means of Detection Times

We developed a new numerical method to estimate bacterial growth parameters by means of detection times generated by different initial counts. The observed detection times are subjected to a transformation involving the (unknown) maximum specific growth rate and the (known) ratios between the different inoculum sizes and the constant detectable level of counts. We present an analysis of variance (ANOVA) protocol based on a theoretical result according to which, if the specific rate used for the transformation is correct, the transformed values are scattered around the same mean irrespective of the original inoculum sizes. That mean, termed the physiological state of the inoculum, , and the maximum specific growth rate, μ, can be estimated by minimizing the variance ratio of the ANOVA procedure. The lag time of the population can be calculated as λ = −ln /μ; i.e. the lag is inversely proportional to the maximum specific growth rate and depends on the initial physiological state of the population. The more accurately the cell number at the detection level is known, the better the estimate for the variance of the lag times of the individual cells.     

Contrasting Effects of Heat Treatment and Incubation Temperature on Germination and Outgrowth of Individual Spores of Nonproteolytic Clostridium botulinum Bacteria

In this study, we determined the effects of incubation temperature and prior heat treatment on the lag-phase kinetics of individual spores of nonproteolytic Clostridium botulinum Eklund 17B. The times to germination (t_germ), one mature cell (t_C1), and two mature cells (t_C2) were measured for individual unheated spores incubated at 8, 10, 15, or 22°C and used to calculate the t_germ, the outgrowth time (t_C1 − t_germ), and the first doubling time (t_C2 − t_C1). Measurements were also made at 22°C of spores that had previously been heated at 80°C for 20 s. For unheated spores, outgrowth made a greater contribution to the duration and variability of the lag phase than germination. Decreasing incubation temperature affected germination less than outgrowth; thus, the proportion of lag associated with germination was less at lower incubation temperatures. Heat treatment at 80°C for 20 s increased the median germination time of surviving spores 16-fold and greatly increased the variability of spore germination times. The shape of the lag-time (t_C1) and outgrowth (t_C1 − t_germ) distributions were the same for unheated spores, but heat treatment altered the shape of the lag-time distribution, so it was no longer homogeneous with the outgrowth distribution. Although heat treatment mainly extended germination, there is also evidence of damage to systems required for outgrowth. However, this damage was quickly repaired and was not evident by the time the cells started to double. The results presented here combined with previous findings show that the stage of lag most affected, and the extent of any effect in terms of duration or variability, differs with both historical treatment and the growth conditions.Clostridium botulinum is a group of four physiologically and phylogenetically distinct anaerobic spore-forming bacteria (known as groups I, II, III, and IV) that produce the highly toxic botulinum neurotoxin (12). The severity of the intoxication, botulism, ensures considerable effort is directed at preventing the growth of this pathogen in food. Nonproteolytic (group II) C. botulinum is one of the two groups most frequently associated with food-borne botulism. It forms heat-resistant spores and can germinate, grow, and produce toxin at 3°C (8); thus, nonproteolytic C. botulinum is a particular concern in mild heat-treated chilled foods (16, 17).Spores formed by pathogens such as C. botulinum are a significant food safety issue since they are able to resist many of the processes, such as cooking, used to kill vegetative cells. Understanding the transformation from a dormant spore to active vegetative cells is an important part of quantifying the risk associated with such organisms. Considerable effort has been targeted at measuring and relating the kinetic responses of populations of C. botulinum to environmental conditions and such data have been used to create predictive models, for example, ComBase (www.combase.cc). Such approaches have made a considerable contribution to ensuring food safety but problems with using population based predictions may arise when an initial inoculum is very small or additional information beyond point values is required. Spores typically contaminate foods at low concentrations so that growth of C. botulinum, when it occurs, is likely to initiate from just a few spores. In these circumstances the distribution of times to growth in packs will reflect the heterogeneity of times to growth from the contaminating individual spores. There is an intrinsic variability between individual spores within a population, and the relationship between population lag and individual lag is complex. Consequently, individual lag times cannot be predicted from population measurements (3). Knowledge of the underlying distribution would allow greater refinement of risk assessments.The lag period between a spore being exposed to conditions suitable for growth and the start of exponential growth will reflect the combined times of germination, emergence, elongation, and first cell division. Currently, very little is known about the variability and duration of these stages and any relationships between them. Measuring the kinetics of spore germination is usually achieved by measuring a population to identify time to percent completion. Such germination curves represent the summation of responses by individual spores. Some authors have measured the biovariability associated with individual spores, but most studies have examined only germination (4-7, 11, 22) and not subsequent outgrowth. More recently, we have used phase-contrast microscopy and image analysis to follow individual spores of nonproteolytic C. botulinum from dormancy, through germination and emergence, to cell division (21, 23). These experiments showed there is very little, or no, relationship between the time spent in each stage by individual spores. We have now extended this work to determine distributions of times for different stages in lag phase as affected by heat treatment and incubation temperature.     

Modeling growth and telomere dynamics in Saccharomyces cerevisiae

A general branching process is proposed to model a population of cells of the yeast Saccharomyces cerevisiae following loss of telomerase. Previously published experimental data indicate that a population of telomerase-deficient cells regain exponential growth after a period of slowing due to critical telomere shortening. The explanation for this phenomenon is that some cells engage telomerase-independent pathways to maintain telomeres that allow them to become “survivors.” Our model takes into account random variation in individual cell cycle times, telomere length, finite replicative lifespan of mother cells, and survivorship. We identify and estimate crucial parameters such as the probability of an individual cell becoming a survivor, and compare our model predictions to experimental data.     

Observing Growth and Division of Large Numbers of Individual Bacteria by Image Analysis

We describe a method that enabled us to observe large numbers of individual bacterial cells during a long period of cell growth and proliferation. We designed a flow chamber in which the cells attached to a transparent solid surface. The flow chamber was mounted on a microscope equipped with a digital camera. The shear force of the flow removed the daughter cells, making it possible to monitor the consecutive divisions of a single cell. In this way, kinetic parameters and their distributions, as well as some physiological characteristics of the bacteria, could be analyzed based on more than 1,000 single-cell observations. The method which we developed enabled us to study the history effect on the distribution of the lag times of single cells.     

Toxoplasma gondii-Vertebrate Cell Interactions. II. The Intracellular Reproductive Phase

SYNOPSIS.
The reproduction of Toxoplasma gondii RH-strain in vertebrate cells was studied in a controlled-environment culture system. The lag period before reproduction and the doubling time of individual parasites were determined using a least-squares linear regression method of analysis which does not artificially constrain the data. In the majority of cases, the time intercept of the linear regression line was either zero, implying the lack of a lag phase before reproduction, or negative, implying the parasite had completed part of its reproductive cycle before entering the host cell. The mean doubling time of T. gondii is 10.9 h in bovine embryo skeletal muscle cells and 8.3 h in HeLa cells. This difference is not significant at the 5% level. The population doubling times of mouse-derived parasites is best described by a gamma distribution.     

Toxoplasma gondii-vertebrate cell interactions. II. The intracellular reproductive phase.

The reproduction of Toxoplasma gondii RH-strain in vertebrate cells was studied in a controlled-environment culture system. The lag period before reproduction and the doubling time of individual parasites were determined using a least-squares linear regression method of analysis which does not artificially constrain the data. In the majority of cases, the time intercept of the linear regression line was either zero, implying the lack of a lag phase before reproduction, or negative, implying the parasite had completed part of its reproductive cycle before entering the host cell. The mean doubling time of T. gondii is 10.9 h in bovine embryo skeletal muscle cells and 8.3 h in HeLa cells. This difference is not significant at the 5% level. The population doubling times of mouse-derived parasites is best described by a gamma distribution.     

Observing growth and division of large numbers of individual bacteria by image analysis

We describe a method that enabled us to observe large numbers of individual bacterial cells during a long period of cell growth and proliferation. We designed a flow chamber in which the cells attached to a transparent solid surface. The flow chamber was mounted on a microscope equipped with a digital camera. The shear force of the flow removed the daughter cells, making it possible to monitor the consecutive divisions of a single cell. In this way, kinetic parameters and their distributions, as well as some physiological characteristics of the bacteria, could be analyzed based on more than 1,000 single-cell observations. The method which we developed enabled us to study the history effect on the distribution of the lag times of single cells.     

Indirect measurement of the lag time distribution of single cells of Listeria innocua in food

The distribution of log counts at a given time during the exponential growth phase of Listeria innocua measured in food samples inoculated with one cell each was applied to estimate the distribution of the single-cell lag times. Three replicate experiments in broth showed that the distribution of the log counts is a linear mapping of the distribution of the detection times measured by optical density. The detection time distribution reflects the lag time distribution but is shifted in time. The log count distribution was applied to estimate the distributions of the lag times in a liquid dairy product and in liver paté after different heat treatments. Two batches of ca. 100 samples of the dairy product were inoculated and heated at 55 degrees C for 45 min or at 62 degrees C for 2 min, and an unheated batch was incubated at 4 degrees C. The final concentration of surviving bacteria was ca. 1 cell per sample. The unheated cells showed the shortest lag times with the smallest variance. The mean and the variance of the lag times of the surviving cells at 62 degrees C were greater than those of the cells treated at 55 degrees C. Three batches of paté samples were heated at 55 degrees C for 25 min, 62 degrees C for 81 s, or 65 degrees C for 20 s. A control batch was inoculated but not heated. All paté samples were incubated at 15 degrees C. The distribution of the lag times of the cells heated at 55 degrees C was not significantly different from that of the unheated cells. However, at the higher temperatures, 62 degrees C and 65 degrees C, the lag duration was longer and its variance greater.     

The use of an automated growth analyser to measure recovery times of single heat-injured Salmonella cells

A new approach to the study of recovery times of single heat-injured Salmonella cells isdescribed. It comprises the generation of a standard heat-injured culture, serial dilution of thisculture to near extinction, inoculation of the serial dilutions across many microtitre plates andmeasurement of the subsequent recovery and growth using an automated turbidometric analyser.Lag times for individual cells were estimated from turbidity data using a model that accuratelyextrapolated the growth curve back to the starting inoculum level. Lag times were comparedusing a number of different commercially available pre-enrichment media. The most typical resultwas a very broad distribution of lag times at the single cell inoculum level, with many values inexcess of 20 h. Even at an inoculum level 10-fold higher, lag times for some injured cells wereestimated to be >10 h. More significantly, it was found that some media recovered more injuredcells than others and vice versa. Between the worst and best media there were as many as 3 log₁₀ cycles difference in the number of cells recoverable. No trends were apparent linkingchoice of medium with performance. The implications of these findings, in relation to traditionaland rapid methodology, are discussed.     

Influence of stress on individual lag time distributions of Listeria monocytogenes

The effects of nine common food industry stresses on the times to the turbidity (T(d)) distribution of Listeria monocytogenes were determined. It was established that the main source of the variability of T(d) for stressed cells was the variability of individual lag times. The distributions of T(d) revealed that there was a noticeable difference in response to the stresses encountered by the L. monocytogenes cells. The applied stresses led to significant changes of the shape, the mean, and the variance of the distributions. The variance of T(d) of wells inoculated with single cells issued from a culture in the exponential growth phase was multiplied by at least 6 and up to 355 for wells inoculated with stressed cells. These results suggest stress-induced variability may be important in determining the reliability of predictive microbiological models.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

A parametric method for cumulative incidence modeling with a new four-parameter log-logistic distribution

Background

The invasion of a new species into an established ecosystem can be directly compared to the steps involved in cancer metastasis. Cancer must grow in a primary site, extravasate and survive in the circulation to then intravasate into target organ (invasive species survival in transport). Cancer cells often lay dormant at their metastatic site for a long period of time (lag period for invasive species) before proliferating (invasive spread). Proliferation in the new site has an impact on the target organ microenvironment (ecological impact) and eventually the human host (biosphere impact).

Results

Tilman has described mathematical equations for the competition between invasive species in a structured habitat. These equations were adapted to study the invasion of cancer cells into the bone marrow microenvironment as a structured habitat. A large proportion of solid tumor metastases are bone metastases, known to usurp hematopoietic stem cells (HSC) homing pathways to establish footholds in the bone marrow. This required accounting for the fact that this is the natural home of hematopoietic stem cells and that they already occupy this structured space. The adapted Tilman model of invasion dynamics is especially valuable for modeling the lag period or dormancy of cancer cells.

Conclusions

The Tilman equations for modeling the invasion of two species into a defined space have been modified to study the invasion of cancer cells into the bone marrow microenvironment. These modified equations allow a more flexible way to model the space competition between the two cell species. The ability to model initial density, metastatic seeding into the bone marrow and growth once the cells are present, and movement of cells out of the bone marrow niche and apoptosis of cells are all aspects of the adapted equations. These equations are currently being applied to clinical data sets for verification and further refinement of the models.     

Recovery of exocellular acid phosphatase activity on Saccharomyces mellis after treatment of the organism with reagents that affect the cell surface

Derepressed cells of Saccharomyces mellis were treated in one of several different ways to either elute or inactivate the exocellular enzyme, acid phosphatase. The enzyme was either (i) eluted from resting cells with 0.5 m KCl plus 0.1% beta-mercaptoethanol, (ii) eluted from exponential phase cells by growing the organism in derepressing media containing 0.5 m KCl, or (iii) inactivated on exponential phase cells by adding sufficient acid or base to growth media to destroy the enzyme but not enough to kill the cells. These treatments did not affect viability. Treated cells were transferred to fresh growth media or some other reaction mixture, and the kinetics of recovery of acid phosphatase activity was studied. In these reaction mixtures, enzyme was synthesized only by actively growing cells. Treated resting cells were indistinguishable from untreated, repressed resting cells in that the organism inoculated into complete growth medium remained in the lag phase for approximately 6 hr before both growth and enzyme synthesis began. Exponential phase derepressed cells treated by method (ii) or (iii) were transferred to fresh medium under conditions that allowed growth to continue. The cells immediately started to manufacture enzyme at a rate greater than normal until the steady-state level was reached, thus demonstrating a feedback control system. Exponential phase repressed cells were also transferred to fresh derepressing media under conditions which sustained growth. Though these cells began to grow immediately, there was a lag before acid phosphatase synthesis began followed by a lengthy inductive period. The length of the period of induction could be correlated with the polyphosphate content of the cells. As the supply of polyphosphate neared exhaustion, the rate of synthesis increased rapidly until it was greater than normal; this differential rate was sustained until the steady-state concentration was reached. When derepressed cells grow in a medium containing 0.5 m KCl, some acid phosphatase activity is found free in the culture fluid and some remains firmly attached to the cells despite the presence of the salt. The bound activity is subject to feedback control, but the steady-state level of this activity on the cells is only one-third that of the acid phosphatase on cells growing in nonsaline media. The extracellular phosphatase is produced at a rate that is several-fold greater than that of the exocellular enzyme in a nonsaline medium. The synthesis of the extracellular enzyme does not seem to be controlled by a feedback mechanism but is produced at a maximal rate as long as the cells are growing.