Comparison of Arrhenius-type and Bêlehrádek-type models for prediction of bacterial growth in foods

D.A. RATKOWSKY, T. ROSS, T.A. WCMEEKIN AND J. OLLEY. 1991. The development of Arrhenius-type ('Schoolfield') and Bêlehrádek-type (square root) models that describe microbial growth rates is briefly described. Both types of model have been advocated for use in predictive microbiology. On the basis of published data sets for the growth of bacteria, the consequences of mathematical transformation of data and the use of invalid stochastic assumptions upon model predictions are demonstrated. Mean square error is shown to be an inappropriate criterion by which to compare the performance of predictive models. The data show that bacterial growth responses such as generation time and lag time become more variable as their mean magnitude increases. The practical consequences of such variability for predictive microbiology are discussed.     

Influence of inoculum preparation on the growth of Penicillium chrysogenum

AIMS: The influence of the spore preparation on subsequent fungal growth of Penicillium chrysogenum was assessed. METHODS AND RESULTS: The influence of four factors [the nature of the diluting solution (physiological water and physiological water added with Tween-80), the age of the sporulating culture (4, 8 and 12 days), the strain (737, 738 and 740) and the inoculum size (102, 103, 104 and 105 spores ml(-1)] on two responses (i.e. the radial growth rate, mu, and the lag time, lambda) was studied using an experimental screening methodology. CONCLUSIONS: The main conclusion was the strong effect of the inoculum size on lambda. In contrast, the diluting solution had no effect on both the experimental responses. In order to obtain the highest growth rates, it is recommended to use 4-day-old sporulating cultures with an inoculum size of 102 spores ml(-1). SIGNIFICANCE AND IMPACT OF THE STUDY: There is a need for standardizing spore preparation in predictive mycology. The screening methodology is a powerful tool to determine the influence of qualitative and quantitative factors on various biological responses and can be applied widely in microbiology.     

Modelling the growth,survival and death of Listeria monocytogenes

In this paper, the predictive microbiology approach has been generalized to the study of growth, survival and death of Listeria monocytogenes. As this micro-organism is involved in food poisoning, its growth, survival and death were studied as functions of low temperatures, NaCl and phenol compounds, in a synthetic medium, by a factorially designed experiment. A significant inactivation of L. monocytogenes was obtained with 20 ppm of phenol and 4% (w/v) NaCl at temperatures from 4 to 12 degrees C. An empirical model is proposed to describe, in a single step, the biomass profile vs studied factors. Thereby, the influence of temperature, NaCl and phenol concentration on L. monocytogenes biomass quantity (0.5-8 log cfu ml(-1)) are presented as a function of storage duration. The comparisons of the proposed model with existing models (Gompertz for growth, vitalistic for survival and death) were performed. The use of a single equation allows the prediction of contamination levels in all experimental conditions without knowledge a priori. The model offers considerable prospects for its use in food microbiology.     

Indices for performance evaluation of predictive models in food microbiology

Two complementary measures are proposed as simple indices of the performance of models in predictive food microbiology. The indices assess the level of confidence one can have in the predictions of the model and whether the model displays any bias which could lead to 'fail-dangerous'predictions. The use of the indices is demonstrated using data collated from independent and published literature. This analysis supports previous reports that evaluation of predictive models by comparison to published microbial growth rate data may be inappropriate because of limitations in that data. The indices may fail to reveal some forms of systematic deviation between observed and predicted behaviour. It is concluded, however, that the indices provide an objective and readily interpreted summary of model performance and may serve as a first step towards the development of an objective and useful definition of the term 'validated model'in predictive food microbiology.     

Bacterial growth laws and their applications

Quantitative empirical relationships between cell composition and growth rate played an important role in the early days of microbiology. Gradually, the focus of the field began to shift from growth physiology to the ever more elaborate molecular mechanisms of regulation employed by the organisms. Advances in systems biology and biotechnology have renewed interest in the physiology of the cell as a whole. Furthermore, gene expression is known to be intimately coupled to the growth state of the cell. Here, we review recent efforts in characterizing such couplings, particularly the quantitative phenomenological approaches exploiting bacterial 'growth laws.' These approaches point toward underlying design principles that can guide the predictive manipulation of cell behavior in the absence of molecular details.     

Analytical solution for a hybrid Logistic-Monod cell growth model in batch and continuous stirred tank reactor culture

Monod and Logistic growth models have been widely used as basic equations to describe cell growth in bioprocess engineering. In the case of the Monod equation, the specific growth rate is governed by a limiting nutrient, with the mathematical form similar to the Michaelis–Menten equation. In the case of the Logistic equation, the specific growth rate is determined by the carrying capacity of the system, which could be growth-inhibiting factors (i.e., toxic chemical accumulation) other than the nutrient level. Both equations have been found valuable to guide us build unstructured kinetic models to analyze the fermentation process and understand cell physiology. In this work, we present a hybrid Logistic-Monod growth model, which accounts for multiple growth-dependent factors including both the limiting nutrient and the carrying capacity of the system. Coupled with substrate consumption and yield coefficient, we present the analytical solutions for this hybrid Logistic-Monod model in both batch and continuous stirred tank reactor (CSTR) culture. Under high biomass yield (Y_x/s) conditions, the analytical solution for this hybrid model is approaching to the Logistic equation; under low biomass yield condition, the analytical solution for this hybrid model converges to the Monod equation. This hybrid Logistic-Monod equation represents the cell growth transition from substrate-limiting condition to growth-inhibiting condition, which could be adopted to accurately describe the multi-phases of cell growth and may facilitate kinetic model construction, bioprocess optimization, and scale-up in industrial biotechnology.     

Evidence for the existence of a secondary pathway for fibril growth during the aggregation of tau

The mechanism of amyloid fibril formation by proteins has been classically described by the nucleation-dependent polymerization (NDP) model, which makes certain predictions regarding the kinetics of fibrillation. All proteins whose aggregation conforms to the NDP model display a t(2) time dependence for their initial reaction profile. However, there are proteins whose aggregation reactions have kinetic signatures of a flat lag phase followed by an exponential rise in fibril mass, which does not conform to the NDP model. Amyloid fibril formation by tau, a microtubule-associated protein whose aggregation to form neurofibrillary tangles is implicated in Alzheimer's disease and other tauopathies, in the presence of inducers such as heparin and fatty acid micelles, has always been traditionally described by a ligand-induced NDP model. In this study, the existence of a secondary pathway for fibril growth during the aggregation of the functional, repeat domain of tau in the presence of heparin has been established. Both kinetic and accessory evidence are provided for the existence of this pathway, which is shown to augment the primary homogeneous nucleation pathway. From the kinetic data, the main secondary pathway that is operative appears to be fibril fragmentation but other pathways such as branching or secondary nucleation may also be operative.     

Development and validation of experimental protocols for use of cardinal models for prediction of microorganism growth in food products

An experimental protocol to validate secondary-model application to foods was suggested. Escherichia coli, Listeria monocytogenes, Bacillus cereus, Clostridium perfringens, and Salmonella were observed in various food categories, such as meat, dairy, egg, or seafood products. The secondary model validated in this study was based on the gamma concept, in which the environmental factors temperature, pH, and water activity (aw) were introduced as individual terms with microbe-dependent parameters, and the effect of foodstuffs on the growth rates of these species was described with a food- and microbe-dependent parameter. This food-oriented approach was carried out by challenge testing, generally at 15 and 10 degrees C for L. monocytogenes, E. coli, B. cereus, and Salmonella and at 25 and 20 degrees C for C. perfringens. About 222 kinetics in foods were generated. The results were compared to simulations generated by existing software, such as PMP. The bias factor was also calculated. The methodology to obtain a food-dependent parameter (fitting step) and therefore to compare results given by models with new independent data (validation step) is discussed in regard to its food safety application. The proposed methods were used within the French national program of predictive microbiology, Sym'Previus, to include challenge test results in the database and to obtain predictive models designed for microbial growth in food products.     

Regulation of cytochrome P-450 CYPIA1 gene expression and proto-oncogene expression by growth factors in primary hepatocytes

The effect of growth factors on the cytochrome P-450 (CYPIA1) gene expression was studied in primary mouse hepatocytes. Of the three growth factors used, i.e. epidermal growth factor (EGF), transforming growth factor alpha (TGF alpha) and insulin, only EGF or TGF alpha completely blocked CYPIA1 expression in the presence of the CYPIA1 inducer 3-methylcholanthrene (3-MC). This repression was not linked to cell cycle progression of the hepatocyte because insulin was active to induce 'early immediate genes' and DNA replication as well as EGF/TGF alpha but failed to suppress CYPIA1 expression. A specific EGF/TGF alpha receptor-mediated function may repress CYPIA1 gene expression and contribute to the acquisition of a xenobiotic drug resistance phenotype.     

Glycosaminoglycan affinity of the complete fibroblast growth factor family

BACKGROUND: Many fibroblast growth factor family proteins (FGFs) bind to the heparan sulfate/heparin (HP) subtypes of sulfated glycosaminoglycans (GAGs), and a few have recently been reported to also interact with chondroitin sulfate (CS), another sulfated GAG subtype. METHODS: To gain additional insight into this interaction, we prepared all currently known FGFs (i.e., FGF1-FGF23) and assessed their affinity for HP, CS-B, CS-D and CS-E. In addition, midkine, hepatocyte growth factor and pleiotrophin were studied as other known HP-binding proteins. RESULTS: We found that members of the FGF19 subfamily (i.e., FGF15, 19, 21 and 23) had little or no affinity for HP; all of the other secretable growth factors tested had strong affinities for HP, as was indicated by the finding that their elution from HP-Sepharose columns required 1.0-1.5 M NaCl. We also found that FGF3, 6, 8 and 22 had strong affinities for CS-E, while FGF5 had a moderate affinity for CS-D. The interactions between FGFs and GAGs thus appear to be more diverse than previously understood. GENERAL SIGNIFICANCE: This is noteworthy, as the differential interactions of these growth factors with GAGs may be key determinants of their specific biological activities.     

A prototype model structure for mixed microbial populations in homogeneous food products

An important factor which has not been included in many models in the field of predictive microbiology is the influence of a background of microflora in a food product. It is however generally known that the growth of a microorganism as a pure culture can be substantially different from its growth in a mixed culture, due to microbial interactions. Because of the importance of these interactions and the lack of suitable modeling techniques in the field of predictive microbiology to describe them, the potential of models in other research fields-namely ecology-to deal with interactions is explored in previous work of the authors. However, a model structure for microbial growth in food products cannot simply be copied from those elaborated in ecology. The structure of a predictive growth model is indeed typical, primarily due to the explicit modeling of a lag phase. The current paper proposes a prototype model structure for growth of mixed microbial populations in homogeneous food products. The model is able to describe a lag phase and reduces to a classical predictive growth model in the special case of single-species growth.     

Challenges in upscaling laboratory studies to ecosystems in soil microbiology research

Soil microbiology has entered into the big data era, but the challenges in bridging laboratory-, field-, and model-based studies of ecosystem functions still remain. Indeed, the limitation of factors in laboratory experiments disregards interactions of a broad range of in situ environmental drivers leading to frequent contradictions between laboratory- and field-based studies, which may consequently mislead model development and projections. Upscaling soil microbiology research from laboratory to ecosystems represents one of the grand challenges facing environmental scientists, but with great potential to inform policymakers toward climate-smart and resource-efficient ecosystems. The upscaling is not only a scale problem, but also requires disentangling functional relationships and processes on each level. We point to three potential reasons for the gaps between laboratory- and field-based studies (i.e., spatiotemporal dynamics, sampling disturbances, and plant–soil–microbial feedbacks), and three key issues of caution when bridging observations and model predictions (i.e., across-scale effect, complex-process coupling, and multi-factor regulation). Field-based studies only cover a limited range of environmental variation that must be supplemented by laboratory and mesocosm manipulative studies when revealing the underlying mechanisms. The knowledge gaps in upscaling soil microbiology from laboratory to ecosystems should motivate interdisciplinary collaboration across experimental, observational, theoretic, and modeling research.     

Predictive modeling of mixed microbial populations in food products: evaluation of two-species models

Predictive microbiology is an emerging research domain in which biological and mathematical knowledge is combined to develop models for the prediction of microbial proliferation in foods. To provide accurate predictions, models must incorporate essential factors controlling microbial growth. Current models often take into account environmental conditions such as temperature, pH and water activity. One factor which has not been included in many models is the influence of a background microflora, which brings along microbial interactions. The present research explores the potential of autonomous continuous-time/two-species models to describe mixed population growth in foods. A set of four basic requirements, which a model should satisfy to be of use for this particular application, is specified. Further, a number of models originating from research fields outside predictive microbiology, but all dealing with interacting species, are evaluated with respect to the formulated model requirements by means of both graphical and analytical techniques. The analysis reveals that of the investigated models, the classical Lotka-Volterra model for two species in competition and several extensions of this model fulfill three of the four requirements. However, none of the models is in agreement with all requirements. Moreover, from the analytical approach, it is clear that the development of a model satisfying all requirements, within a framework of two autonomous differential equations, is not straightforward. Therefore, a novel prototype model structure, extending the Lotka-Volterra model with two differential equations describing two additional state variables, is proposed to describe mixed microbial populations in foods.     

Interaction of circulating cell-derived and plasma growth factors in stimulating cultured smooth muscle cell replication

Multiple growth factors that circulate in plasma have been shown to stimulate cellular growth in vitro. The plasma growth factors appear to stimulate DNA synthesis in cultured fibroblasts only after prior exposure of cell growth factors derived from circulating cell types, such as platelets and macrophages. The purpose of these studies was to investigate the role of the plasma growth factors in stimulating smooth muscle cell replication following exposure to platelet-derived growth factor (PDGF). Following transient exposure to PDGF, insulin stimulated smooth muscle cell replication but only when supraphysiologic concentrations were used (i.e., greater than 1.0 μg/ml). Somatomedin-C (Sm-C), in contrast, was found to stimulate a 320% increase in [³H]thymidine incorporation when concentrations that are present in extracellular fluids were used (i.e., 0.5–10 ng/ml). Epidermal growth factor (EGF), an important mitogen for multiple cell types, caused a 70% increase in [³H]thymidine incorporation when added to quiescent cells following PDGF exposure, and EGF caused a substantial increase in the absolute level of [³H]thymidine incorporation when coincubated with Sm-C. When EGF (1 ng/ml) was incubated simultaneously with concentrations of Sm-C between 1 and 10 ng/ml plus Sm-C-deficient plasma, maximal [³H]thymidine incorporation was 2.1-fold greater in the presence of EGF. In contrast, insulin (20 ng/ml), when coincubated with Sm-C under similar conditions, had no enhancing effect on the cellular response to Sm-C. None of the plasma factors tested was an effective stimultant of replication when incubated either in serum-free medium or in the presence of Sm-C-deficient plasma without prior PDGF exposure. Hydrocortisone was shown to inhibit smooth muscle cell replication in concentrations between 10^?7 and 10^?5M. In summary, multiple plasma growth factors can stimulate the smooth muscle cell replication, and Sm-C appears to be most effective of those tested. Insulin and EGF appear to work by different mechanisms; that is, EGF can facilitate the cellular response to Sm-C, whereas insulin is effective only at supraphysiologic concentrations at which it will directly bind to Sm-C receptors.     

Growth Kinetics of Suspended Microbial Cells: From Single-Substrate-Controlled Growth to Mixed-Substrate Kinetics

Growth kinetics, i.e., the relationship between specific growth rate and the concentration of a substrate, is one of the basic tools in microbiology. However, despite more than half a century of research, many fundamental questions about the validity and application of growth kinetics as observed in the laboratory to environmental growth conditions are still unanswered. For pure cultures growing with single substrates, enormous inconsistencies exist in the growth kinetic data reported. The low quality of experimental data has so far hampered the comparison and validation of the different growth models proposed, and only recently have data collected from nutrient-controlled chemostat cultures allowed us to compare different kinetic models on a statistical basis. The problems are mainly due to (i) the analytical difficulty in measuring substrates at growth-controlling concentrations and (ii) the fact that during a kinetic experiment, particularly in batch systems, microorganisms alter their kinetic properties because of adaptation to the changing environment. For example, for Escherichia coli growing with glucose, a physiological long-term adaptation results in a change in K_S for glucose from some 5 mg liter⁻¹ to ca. 30 μg liter⁻¹. The data suggest that a dilemma exists, namely, that either “intrinsic” K_S (under substrate-controlled conditions in chemostat culture) or μ_max (under substrate-excess conditions in batch culture) can be measured but both cannot be determined at the same time. The above-described conventional growth kinetics derived from single-substrate-controlled laboratory experiments have invariably been used for describing both growth and substrate utilization in ecosystems. However, in nature, microbial cells are exposed to a wide spectrum of potential substrates, many of which they utilize simultaneously (in particular carbon sources). The kinetic data available to date for growth of pure cultures in carbon-controlled continuous culture with defined mixtures of two or more carbon sources (including pollutants) clearly demonstrate that simultaneous utilization results in lowered residual steady-state concentrations of all substrates. This should result in a competitive advantage of a cell capable of mixed-substrate growth because it can grow much faster at low substrate concentrations than one would expect from single-substrate kinetics. Additionally, the relevance of the kinetic principles obtained from defined culture systems with single, mixed, or multicomponent substrates to the kinetics of pollutant degradation as it occurs in the presence of alternative carbon sources in complex environmental systems is discussed. The presented overview indicates that many of the environmentally relevant apects in growth kinetics are still waiting to be discovered, established, and exploited.     

Nutrient growth conditions for Oscillatoria spp. and diatoms in two Norwegian lakes

The growth conditions and development of Asterionella formosa Hass., Tabellaria fenestrata (Lyngb.) Kütz., Fragilaria crotonensis Kitt., and Oscillatoria spp. have been studied in Lake Vansjø and Lake Mjøsa for many years. Some batch culture experiments have been performed with dilute phytoplankton populations in filtered water samples to support the field data. The results indicate that the concentrations of phosphate, nitrate, silicate or chelators/Fe were important factors regulating the spatial and temporal distribution of the populations. Based on the assumption that the growth rates were strongly dependent on the external nutrient concentration, S, the growth conditions were divided into three types. The nutrient concentration may be too low to support growth, i.e. the growth rate k ≤ 0, or the population is nutrient growth limited (0 < k < km) or not growth limited, i.e. the population has maximum growth rate (km). The nutrient concentration may be population density dependent, i.e. resource competition may occur, or density independent separately or in combination. The growth strategies of the populations in the two lakes were partly explained by studying the nutrient growth conditions of the populations.     

Growth kinetics for shelf-life prediction: Theory and practice

Summary Predictive microbiology can be used to determine and predict the shelf-life of perishable foods under commercial distribution conditions based on microbial growth kinetics. This paper presents general microbial growth kinetics with the Monod model and the Gompertz function. Additional models are given to describe effects of food composition (e. g.a _w) and environmental conditions (e.g. temperature, gas atmosphere) as well as their interaction on the growth kinetic parameters (lag time and specific growth rate). These models can be used to predict the time to reach a critical level under any constant conditions within the range tested. A combination of microbial kinetics with an engineering accumulation approach can be used to predict the final microbial level in a food, or the loss of shelf-life, for any known time-temperature sequence, if there is no history effect or the history effect is negligible. A time-temperature indicator, could be used for predicting the remaining shelf-life of perishable foods under any distribution condition based on microbial growth kinetics.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Competition between two microbial populations in a nonmixed environment: effect of cell random motility

In a nonmixed environment, bacterial population growth can be influenced significantly by cell motility properties as well as by growth kinetic properties. Therefore, in a situation of competition between two bacterial populations for a single chemical nutrient in a nonmixed environment, the outcome may depend upon the respective cell motility properties. In this article, the authors have presented a simple mathematical model for competitive growth of two randomly motile (i.e., possessing no chemotactic behavior) populations in a finite nonmixed region. An understanding of the behavior of this model should provide insight into the behavior of a number of common microbial competition problems. Analysis of this model yields the following results: (1) There may be as many as three possible non-trivial steady-state (or long-time) configurations: when species 1 survives, species 2 dies out; when species 2 survives, species 1 dies out; and species 1 and species 2 coexist. (2) The coexistence state can exist even though one species possesses a smaller intrinsic growth rate constant at all nutrient concentrations, if that same species is sufficiently less motile than the other species. (3) In fact, the species with the smaller maximum specific growth rate may grow to a larger population than the other. (4) The possibility of coexistence can be decided essentially from the results for single population growth.     

Allometry of resource capture in colonial cnidarians and constraints on modular growth

1. Indeterminacy in growth of colonial organisms, such as corals, is commonly attributed to their modular construction which frees the colony from the allometric constraints that limit the size of single modules. However, as a colony grows, there may be a decrease in resource availability to interior modules because of active depletion and/or passive deflection by modules on the exterior. The effects of 'self-shading' on resource capture in modular animals are modelled using a simple allometric growth function.
2. The model assumes that resource capture by a module scales as an exponent ( γ ) of colony size (i.e. number of modules). Data taken from the literature indicate that model values of γ for light and prey capture range from – 0·80 to – 1·16 for branching and encrusting corals. Module-specific rates of resource use (i.e. metabolism) are less affected by colony size. Therefore, as a colony grows, net resource state eventually reaches zero, making further growth unsustainable or determinate.
3. The model also predicts an inverse relationship between module size and colony size such as that observed in Caribbean corals. This negative correlation results from the additive effects of module size and colony size on the degree of self-shading.
4. Resource capture is affected by growth form and flow regime, and the interaction between them can account for some of the morphological variation in corals and other colonial suspension feeders.