Cell division theory and individual-based modeling of microbial lag: part II. Modeling lag phenomena induced by temperature shifts

This paper is the second in a series of two, and studies microbial lag in cell number and/or biomass measurements caused by temperature changes with an individual-based modeling approach. For this purpose, the theory of cell division, as discussed in the first part of this series of research papers, was implemented in the individual-based modeling framework BacSim. Simulations of this model are compared with experimental data of Escherichia coli, growing in an aerated, glucose-rich medium and subjected to sudden temperature shifts. The premise of a constant cell volume under changing temperature conditions predicts no lag in cell numbers after the shift, in contrast to the experimental observations. Based on literature research, two biological mechanisms that could be responsible for the observed lag phenomena are proposed. The first assumes that the average cell volume depends on temperature while the second assumes that a lag in biomass growth occurs after the temperature shift. For a lag in cell number caused by an increased average cell volume, the cell biomass always increases at the maximal rate. Therefore, cells are evidently not stressed and do not have to adapt to the new conditions, as opposed to a lag in biomass growth. Implementation and simulation of both mechanisms are found to describe the experimental observations equally well. Therefore, further research is needed to distinguish between the two mechanisms. This can be done by observing, in addition to cell numbers, a measure for the average cell volumes. In conclusion, the individual-based modeling approach is a good methodology to investigate and test biological theories and assumptions. Also, based on the simulations, suggestions for further experimental observations can be made.     

Predictive modelling of the microbial lag phase: a review

This paper summarises recent trends in predictive modelling of microbial lag phenomena. The lag phase is approached from both a qualitative and a quantitative point of view. First, a definition of lag and an analysis of the prevailing measuring techniques for the determination of lag time is presented. Furthermore, based on experimental results presented in literature, factors influencing the lag phase are discussed. Major modelling approaches concerning lag phase estimation are critically assessed. In predictive microbiology, a two-step modelling approach is used. Primary models describe the evolution of microbial numbers with time and can be subdivided into deterministic and stochastic models. Primary deterministic models, e.g., Baranyi and Roberts [Int. J. Food Microbiol. 23 (1994) 277], Hills and Wright [J. Theor. Biol. 168 (1994) 31] and McKellar [Int. J. Food Microbiol. 36 (1997) 179], describe the evolution of microorganisms, using one single (deterministic) set of model parameters. In stochastic models, e.g., Buchanan et al. [Food Microbiol. 14 (1997) 313], Baranyi [J. Theor. Biol. 192 (1998) 403] and McKellar [J. Appl. Microbiol. 90 (2001) 407], the model parameters are distributed or random variables. Secondary models describe the relation between primary model parameters and influencing factors (e.g., environmental conditions). This survey mainly focuses on the influence of temperature and culture history on the lag phase during growth of bacteria.     

Effect of chemicals on the microbial evolution in foods

In contrast with most chemical hazardous compounds, the concentration of food pathogens changes during processing, storage, and meal preparation, making it difficult to estimate the number of microorganisms or the concentration of their toxins at the moment of ingestion by the consumer. These changes are attributed to microbial proliferation, survival, and/or inactivation and must be considered when exposure to a microbial hazard is assessed. The number of microorganisms can also change as a result of physical removal, mixing of food ingredients, partitioning of a food product, or cross-contamination (M. J. Nauta. 2002. Int. J. Food Microbiol. 73:297-304). Predictive microbiology, i.e., relating these microbial evolutionary patterns to environmental conditions, can therefore be considered a useful tool for microbial risk assessment, especially in the exposure assessment step. During the early development of the field (late 1980s and early 1990s), almost all research was focused on the modeling of microbial growth over time and the influence of temperature on this growth. Later, modeling of the influence of other intrinsic and extrinsic parameters garnered attention. Recently, more attention has been given to modeling of the effects of chemicals on microbial inactivation and survival. This article is an overview of different applied strategies for modeling the effect of chemical compounds on microbial populations. Various approaches for modeling chemical growth inhibition, the growth-no growth interface, and microbial inactivation by chemicals are reviewed.     

A model describing the relationship between regrowth lag time and mild temperature increase for Listeria monocytogenes

In order to comply with the consumer demand for ready-to-eat and look 'fresh' products, mild heat treatment will be used more and more in the agrofood industry. Nonetheless there is no tool to define the most appropriate mild heat treatment. In order to build this tool, it is necessary to study and describe the response of a bacterial population to a mild increase in temperature, from the dynamic point of view. The response to a mild increase in temperature, defined by stress duration and temperature, consisted in a mortality phase followed by the lag time of the survivors and their exponential growth. The effect of the mild increase in temperature on the mortality phase was described in a previous paper (Bréand et al., Int. J. Food Microbiol., in press). The effect of the stress duration on the lag was presented in a previous paper (Bréand et al., Int. J. Food Microbiol. 38 (1997) 157-167). In particular, the biphasic relationship between the lag and the stress duration was observed and modelled with a four parameter nonlinear model: the primary model (Bréand et al., Int. J. Food Microbiol. 38 (1997) 157-167). The study presented in this paper deals with the effect of the stress temperature on the biphasic relationship between the lag time and the stress duration. The secondary models describing the effect of the stress temperature on this biphasic relationship, were empirically built from our experimental data concerning Listeria monocytogenes. This work pointed out that the higher the stress temperature, the narrower the range of stress duration for which the lag time increased. Since the primary and the secondary models of the lag time were available, the global model describing the effect of the mild increase duration and temperature directly on the lag was fitted. This model allowed an improvement of the parameter estimator precision. The potential contribution in mild heat treatment optimization of this global model and the one built for the mortality phase (Bréand et al., Int. J. Food Microbiol., in press) is discussed.     

Viable count estimates of lag time responses for Salmonella typhimurium M48 subjected to abrupt osmotic shifts

Generally, relative lag times (RLT; lag time divided by generation time) become extended as conditions become less favourable for growth. Mellefont et al. (2003, 2004) [Mellefont, L.A., McMeekin, T.A., Ross, T., 2003. The effect of abrupt osmotic shifts on the lag phase duration of foodborne bacteria. Int. J. Food Microbiol. 83(3), 281-293; Mellefont, L.A., McMeekin, T.A., Ross, T., 2004. The effect of abrupt osmotic shifts on the lag phase duration of physiologically distinct populations of Salmonella typhimurium. Int. J. Food Microbiol. 92, 111-120] reported that abrupt osmotic shifts of Salmonella typhimurium M48 from optimal to low aw led to unexpectedly small RLTs at low aw. In this study, RLTs resulting from similar osmotic shifts were estimated by viable count (VC) and compared to turbidimetric estimates to test the hypothesis that the 'downturn' in RLT is an artefact of the turbidimetric technique used. No 'downturn' in RLT was observed with VC data and RLTs increased with increasing magnitude of osmotic shift. Anomalous turbidimetric estimates of lag time at low aw were confirmed as the likely source of the 'downturn' in RLT. The abrupt osmotic shifts resulted in a complex pattern of microbial population behaviour. Immediately after transfer from optimal aw to low aw, inactivation of a portion of the population occurred for all the conditions tested. The degree of inactivation became progressively larger with larger shifts in aw. The initial decline in population was followed by a period during which no change in numbers occurred, followed by growth that appeared, in most cases, to be exponential. At the lowest aws tested (< or =0.954), the growth response after the initial decline was at a rate slower than that of exponential phase growth. Due to the use of non-selective media containing pyruvate (to eliminate oxygen radicals), the observed patterns of inactivation, lag and regrowth at most aw conditions are unlikely to result from a temporary loss of culturability, but may represent inactivation of a portion of the population.     

On the design of optimal dynamic experiments for parameter estimation of a Ratkowsky-type growth kinetics at suboptimal temperatures

It is generally known that accurate model building, i.e., proper model structure selection and reliable parameter estimation, constitutes an essential matter in the field of predictive microbiology, in particular, when integrating these predictive models in food safety systems. In this context, Versyck et al. (1999) have introduced the methodology of optimal experimental design techniques for parameter estimation within the field. Optimal experimental design focuses on the development of optimal input profiles such that the resulting rich (i.e., highly informative) experimental data enable unique model parameter estimation. As a case study, Versyck et al. (1999) [Versyck, K., Bernaerts, K., Geeraerd, A.H., Van Impe, J.F., 1999. Introducing optimal experimental design in predictive modeling: a motivating example. Int. J. Food Microbiol., 51(1), 39-51] have elaborated the estimation of Bigelow inactivation kinetics parameters (in a numerical way). Opposed to the classic (static) experimental approach in predictive modelling, an optimal dynamic experimental setup is presented. In this paper, the methodology of optimal experimental design or parameter estimation is applied to obtain uncorrelated estimates of the square root model parameters [Ratkowsky, D.A., Olley, J., McMeekin, T.A., Ball, A., 1982. Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149, 1-5] describing the effect of suboptimal growth temperatures on the maximum specific growth rate of microorganisms. These estimates are the direct result of fitting a primary growth model to cell density measurements as a function of time. Apart from the design of an optimal time-varying temperature profile based on a sensitivity study of the model output, an important contribution of this publication is a first experimental validation of this innovative dynamic experimental approach for uncorrelated parameter identification. An optimal step temperature profile, within the range of model validity and practical feasibility, is developed for Escherichia coli K12 and successfully applied in practice. The presented experimental validation result illustrates the large potential of the dynamic experimental approach in the context of uncorrelated parameter estimation. Based on the experimental validation result, additional remarks are formulated related to future research in the field of optimal experimental design.     

食源性致病菌生长延滞期建模的研究进展

基于预测微生物学理论,准确预测食源性致病菌的数量及生长对降低食品安全风险具有重大意义。本文针对近年来国内外食源性致病菌生长延滞期的建模工作,从食源性致病菌延滞期的测定方法和建模方法两方面进行综述。通过分析现有延滞期建模的相关研究,指出目前延滞期建模研究中存在的问题。建议未来应基于微生物生长机理明确延滞期定义,进一步根据延滞期定义改进、创新延滞期测定方法及建模方法,同时量化分析延滞期影响因素并整合延滞期建模。     

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.     

Predictive microbiology models vs. modeling microbial growth within Listeria monocytogenes risk assessment: what parameters matter and why

Predictive microbiology models are essential tools to model bacterial growth in quantitative microbial risk assessments. Various predictive microbiology models and sets of parameters are available: it is of interest to understand the consequences of the choice of the growth model on the risk assessment outputs. Thus, an exercise was conducted to explore the impact of the use of several published models to predict Listeria monocytogenes growth during food storage in a product that permits growth. Results underline a gap between the most studied factors in predictive microbiology modeling (lag, growth rate) and the most influential parameters on the estimated risk of listeriosis in this scenario (maximum population density, bacterial competition). The mathematical properties of an exponential dose-response model for Listeria accounts for the fact that the mean number of bacteria per serving and, as a consequence, the highest achievable concentrations in the product under study, has a strong influence on the estimated expected number of listeriosis cases in this context.     

Towards a novel class of predictive microbial growth models

Food safety and quality are influenced by the presence (and possible proliferation) of pathogenic and spoilage microorganisms during the life cycle of the product (i.e., from the raw ingredients at the start of the production process until the moment of consumption). In order to simulate and predict microbial evolution in foods, mathematical models are developed in the field of predictive microbiology. In general, microbial growth is a self-limiting process, principally due to either (i) the exhaustion of one of the essential nutrients, and/or (ii) the accumulation of toxic products that inhibit growth. Nowadays, most mathematical models used in predictive microbiology do not explicitly incorporate this basic microbial knowledge. In this paper, a novel class of microbial growth models is proposed. In contrast with the currently used logistic type models, e.g., the model of Baranyi and Roberts [Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277–294], the novel model class explicitly incorporates nutrient exhaustion and/or metabolic waste product effects. As such, this novel model prototype constitutes an elementary building block to be extended in a natural way towards, e.g., microbial interactions in co-cultures (mediated by metabolic products) and microbial growth in structured foods (influenced by, e.g., local substrate concentrations). While under certain conditions the mathematical equivalence with classical logistic type models is clear and results in equal fitting capacities and parameter estimation quality (see Poschet et al. [Poschet, F., Vereecken, K.M., Geeraerd, A.H., Nicolaï, B.M., Van Impe, J.F., 2004. Analysis of a novel class of predictive microbial growth models and application to co-culture growth. International Journal of Food Microbiology, this issue] for a more elaborated analysis in this respect), the biological interpretability and extendability represent the main added value.     

A quantitative approach for studying the effect of heat treatment conditions on resistance and recovery of Bacillus cereus spores.

When a bacterial population undergoes an unfavourable transient increase in temperature, a death phase followed by a lag and growth phase are observed for the surviving and cultivable population. The lag phase is of great interest in regard to food safety, but for bacterial spores, very few studies have been carried out on the evolution of lag time versus heat treatment duration. The experiments monitored on spores of two strains of Bacillus cereus showed a biphasic evolution of the lag time for germination of stressed spores with increase in heat treatment duration, at 90 degrees C, 95 degrees C and 100 degrees C and for different recovery conditions in laboratory medium. Initially, the lag time increased with the duration of the thermal stress up to an optimum. Subsequently, the lag time decreased, with increasing stress duration, to a threshold. The evolution of the lag time to germination of surviving spores, under the range of temperature tested, was similar to the evolution of the lag time to growth after mild-temperature treatments observed with vegetative cells of Escherichia coli and Listeria monocytogenes by Breand et al. (1997) [Breand, S., Fardel, G., Flandrois, J.-P., Rosso, L., Tomassone, R., 1997. A model describing the relationship between lag time and mild temperature increase duration. Int. J. Food Microbiol. 38, 157-167]. The mathematical model established from these investigations could also be applied in the case of heat-stressed spores. Moreover, it was observed that there was a significant linear relationship between the D-value and the treatment duration that gave the maximum lag time.     

Engineering modeling frameworks for microbial food safety at various scales

The landscape of mathematical model-based understanding of microbial food safety is wide and deep, covering interdisciplinary fields of food science, microbiology, physics, and engineering. With rapidly growing interest in such model-based approaches that increasingly include more fundamental mechanisms of microbial processes, there is a need to build a general framework that steers this evolutionary process by synthesizing literature spread over many disciplines. The framework proposed here shows four interconnected, complementary levels of microbial food processes covering sub-cellular scale, microbial population scale, food scale, and human population scale (risk). A continuum of completely mechanistic to completely empirical models, widely-used and emerging, are integrated into the framework; well-known predictive microbiology modeling being a part of this spectrum. The framework emphasizes fundamentals-based approaches that should get enriched over time, such as the basic building blocks of microbial population scale processes (attachment, migration, growth, death/inactivation and communication) and of food processes (e.g., heat and moisture transfer). A spectrum of models are included, for example, microbial population modeling covers traditional predictive microbiology models to individual-based models and cellular automata. The models are shown in sufficient quantitative detail to make obvious their coupling, or their integration over various levels. Guidelines to combine sub-processes over various spatial and time scales into a complete interdisciplinary and multiphysics model (i.e., a system) are provided, covering microbial growth/inactivation/transport and physical processes such as fluid flow and heat transfer. As food safety becomes increasingly predictive at various scales, this synthesis should provide its roadmap. This big picture and framework should be futuristic in driving novel research and educational approaches.     

Modeling microbial competition in food: application to the behavior of Listeria monocytogenes and lactic acid flora in pork meat products

Competition between background microflora and microbial pathogens raises questions about the application of predictive microbiology in situ, i.e., in non-sterile naturally contaminated foods. In this article, we present a review of the models developed in predictive microbiology to describe interactions between microflora in foods, with a special focus on two approaches: one based on the Jameson effect (simultaneous deceleration of all microbial populations) and one based on the Lotka-Volterra competition model. As an illustration of the potential of these models, we propose various modeling examples in estimation and in prediction of microbial growth curves, all related to the behavior of Listeria monocytogenes with lactic acid bacteria in three pork meat products (fresh pork meat and two types of diced bacon).     

Measurements and predictions of growth for Listeria monocytogenes and Salmonella during fluctuating temperature II. Rapidly changing temperatures

Growth of Listeria monocytogenes and Salmonella was examined during various rates of increase and decrease in temperature from and to the minimum for growth. Growth was little affected by even the most rapid changes and injury or lag was not observed. Subsequent investigations of growth during periods of rapid variation in temperature from and to temperatures below the growth minimum again had little effect and growth was satisfactorily predicted using the dynamic model of Baranyi and Roberts [Int. J. Food Microbiol. 23 (1994) 277] in conjunction with the data of Food Micromodel.     

Stochastic modelling of individual cell growth using flow chamber microscopy images

In this paper, we analysed individual cell growth images obtained by flow chamber microscopy system. We used replicate flow chamber experiment data as reported by [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] involving both unstressed and heat shocked Listeria innocua cells. After observing the kinetics of a large number of cells, we propose a new stochastic model for their individual growth. By comparing our model with other existing models in the literature, we demonstrate that ours can accurately describe the growth of both stressed and unstressed cells. Our results indicate that the lag period, in terms of cell division, coincides dominantly with a lag period in terms of cell size. We also reveal various connections between cell length, lag time and cell division models. Finally, we present the results of our investigation on the effect of the duration of sublethal heat shock on the found growth properties.     

肉品微生物生长预测模型研究进展

预测微生物学是一门利用现有的数据去预测未来发展趋势的新兴学科,这种方式可以对实际生产和流通过程进行监控.本文介绍了肉品微生物生长预测模型构建的基本过程、模型选择原则和评价方法.肉品介质是一个复杂的环境体系,在微生物生长预测模型预测过程中影响因素较多,应该根据具体的情况建立或推导出合适的模型来预测肉品中微生物的生长.     

Analysis of the lag phase to exponential growth transition by incorporating inoculum characteristics

During the last decade, individual-based modelling (IbM) has proven to be a valuable tool for modelling and studying microbial dynamics. As each individual is considered as an independent entity with its own characteristics, IbM enables the study of microbial dynamics and the inherent variability and heterogeneity. IbM simulations and (single-cell) experimental research form the basis to unravel individual cell characteristics underlying population dynamics. In this study, the IbM framework MICRODIMS, i.e., MICRObial Dynamics Individual-based Model/Simulator, is used to investigate the system dynamics (with respect to the model and the system modelled). First, the impact of the time resolution on the simulation accuracy is discussed. Second, the effect of the inoculum state and size on emerging individual dynamics, such as individual mass, individual age and individual generation time distribution dynamics, is studied. The distributions of individual characteristics are more informative during the lag phase and the transition to the exponential growth phase than during the exponential phase. The first generation time distributions are strongly influenced by the inoculum state. All inocula with a pronounced heterogeneity, except the inocula starting from a uniform distribution, exhibit commonly observed microbial behaviour, like a more spread first generation time distribution compared to following generations and a fast stabilisation of biomass and age distributions.     

Predictive microbiology: providing a knowledge-based framework for change management

This contribution considers predictive microbiology in the context of the Food Micro 2002 theme, "Microbial adaptation to changing environments". To provide a reference point, the state of food microbiology knowledge in the mid-1970s is selected and from that time, the impact of social and demographic changes on microbial food safety is traced. A short chronology of the history of predictive microbiology provides context to discuss its relation to and interactions with hazard analysis critical control point (HACCP) and risk assessment. The need to take account of the implications of microbial adaptability and variable population responses is couched in terms of the dichotomy between classical versus quantal microbiology introduced by Bridson and Gould [Lett. Appl. Microbiol. 30 (2000) 95]. The role of population response patterns and models as guides to underlying physiological processes draws attention to the value of predictive models in development of novel methods of food preservation. It also draws attention to the paradox facing today's food industry that is required to balance the "clean, green" aspirations of consumers with the risk, to safety or shelf life, of removing traditional barriers to microbial development. This part of the discussion is dominated by consideration of models and responses that lead to stasis and inactivation of microbial populations. This highlights the consequence of change on predictive modelling where the need is now to develop interface and non-thermal death models to deal with pathogens that have low infective doses for general and/or susceptible populations in the context of minimal preservation treatments. The challenge is to demonstrate the validity of such models and to develop applications of benefit to the food industry and consumers as was achieved with growth models to predict shelf life and the hygienic equivalence of food processing operations.