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

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

Adaptation of soil microbial communities to temperature: comparison of fungi and bacteria in a laboratory experiment

Temperature not only has direct effects on microbial activity, but can also affect activity indirectly by changing the temperature dependency of the community. This would result in communities performing better over time in response to increased temperatures. We have for the first time studied the effect of soil temperature (5–50 °C) on the community adaptation of both bacterial (leucine incorporation) and fungal growth (acetate-in-ergosterol incorporation). Growth at different temperatures was estimated after about a month using a short-term assay to avoid confounding the effects of temperature on substrate availability. Before the experiment started, fungal and bacterial growth was optimal around 30 °C. Increasing soil temperature above this resulted in an increase in the optimum for bacterial growth, correlated to soil temperature, with parallel shifts in the total response curve. Below the optimum, soil temperature had only minor effects, although lower temperatures selected for communities growing better at the lowest temperature. Fungi were affected in the same way as bacteria, with large shifts in temperature tolerance at soil temperatures above that of optimum for growth. A simplified technique, only comparing growth at two contrasting temperatures, gave similar results as using a complete temperature curve, allowing for large scale measurements also in field situations with small differences in temperature.     

Modeling Surface Growth of Escherichia coli on Agar Plates

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

Modeling surface growth of Escherichia coli on agar plates

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

Comparison of temperature effects on soil respiration and bacterial and fungal growth rates

Temperature is an important factor regulating microbial activity and shaping the soil microbial community. Little is known, however, on how temperature affects the most important groups of the soil microorganisms, the bacteria and the fungi, in situ. We have therefore measured the instantaneous total activity (respiration rate), bacterial activity (growth rate as thymidine incorporation rate) and fungal activity (growth rate as acetate-in-ergosterol incorporation rate) in soil at different temperatures (0-45 degrees C). Two soils were compared: one was an agricultural soil low in organic matter and with high pH, and the other was a forest humus soil with high organic matter content and low pH. Fungal and bacterial growth rates had optimum temperatures around 25-30 degrees C, while at higher temperatures lower values were found. This decrease was more drastic for fungi than for bacteria, resulting in an increase in the ratio of bacterial to fungal growth rate at higher temperatures. A tendency towards the opposite effect was observed at low temperatures, indicating that fungi were more adapted to low-temperature conditions than bacteria. The temperature dependence of all three activities was well modelled by the square root (Ratkowsky) model below the optimum temperature for fungal and bacterial growth. The respiration rate increased over almost the whole temperature range, showing the highest value at around 45 degrees C. Thus, at temperatures above 30 degrees C there was an uncoupling between the instantaneous respiration rate and bacterial and fungal activity. At these high temperatures, the respiration rate closely followed the Arrhenius temperature relationship.     

鲜切生菜中沙门氏菌二级生长预测模型的建立

以鲜切生菜为研究对象,比较了修正的Gompertz、Gompertz、Logistic和MMF 4种一级模型对不同温度下鲜切生菜中沙门氏菌生长曲线的拟合情况,发现在36℃、20℃和10℃时,修正的Gompertz模型均为最佳的拟合模型,4℃时沙门氏菌生长受到抑制,对失活/存活曲线进行"镜像化"处理后发现拟合程度相对较低,相关系数为0.962 7,故未用于二级模型中;采用其他温度下的修正的Gompertz模型中的最大比生长速率作为二级模型的响应值,建立平方根二级模型;准确因子和偏差因子对二级模型的准确性验证结果表明,两者均接近1.0,说明所建立的二级模型用于预测鲜切生菜中沙门氏菌生长情况。本研究为鲜切生菜的微生物安全控制提供科学依据。     

Environmental variability counteracts priority effects to facilitate species coexistence: evidence from nectar microbes

The order of species arrival during community assembly can greatly affect species coexistence, but the strength of these effects, known as priority effects, appears highly variable across species and ecosystems. Furthermore, the causes of this variation remain unclear despite their fundamental importance in understanding species coexistence. Here, we show that one potential cause is environmental variability. In laboratory experiments using nectar-inhabiting microorganisms as a model system, we manipulated spatial and temporal variability of temperature, and examined consequences for priority effects. If species arrived sequentially, multiple species coexisted under variable temperature, but not under constant temperature. Temperature variability prevented extinction of late-arriving species that would have been excluded owing to priority effects if temperature had been constant. By contrast, if species arrived simultaneously, species coexisted under both variable and constant temperatures. We propose possible mechanisms underlying these results using a mathematical model that incorporates contrasting effects of microbial species on nectar pH and amino acids. Overall, our findings suggest that understanding consequences of priority effects for species coexistence requires explicit consideration of environmental variability.     

An annual cycle,at constant temperature,in a model sandy beach. II. Microbial biology

A model sandy beach successfully mimicking the natural environment has been operated under controlled conditions at a constant temperature (5°C, representative of British winter coastal sea-water temperatures) for a year. During this time the bacterial population of the sand and inundating sea water has been enumerated. Bacterial numbers were highest near the high and low water marks and had a minimum in between on the beach slope; they decreased with depth in the sand profile and showed little variation with time. A number of the bacteria were isolated from the sand system and a preliminary classification attempted. The results are discussed in relation to the meagre literature pertaining to the microbial biology of natural marine sandy beaches.     

Relationship between inactivation kinetics of a Listeria monocytogenes suspension by chlorine and its chlorine demand

AIMS: Chlorine demand by Listeria monocytogenes cells and inactivation of L. monocytogenes by chlorine (0.6-1.0 mg l(-1)) at different temperatures (4, 20 and 30 degrees C) have been investigated in a batch reactor. METHODS AND RESULTS: Chlorine demand depended on the microbial concentration and was independent on the initial chlorine concentration and temperature. Chlorine decay was modelled by the addition of two first-order decay equations. Inactivation of L. monocytogenes by chlorine depended on the initial microbial concentration, initial chlorine concentration and temperature. A mathematical model based on a biphasic inactivation properly described survival curves of L. monocytogenes and a tertiary model was developed that satisfactorily predicted the inactivation of L. monocytogenes by different concentrations of initial chlorine at different temperatures. CONCLUSIONS: Both available chlorine decay and inactivation of L. monocytogenes by chlorine were biphasic and can be modelled by a two-term exponential model. SIGNIFICANCE AND IMPACT OF THE STUDY: The biphasic nature of survival curves of L. monocytogenes did not reflect the effect of a change of available chlorine concentration during the treatment. The microbial inactivation was caused by successive reactions that occur after the consumption of the chlorine by the bacterial cell components.     

Comparing temperature sensitivity of bacterial growth in Antarctic marine water and soil

The western Antarctic Peninsula is an extreme low temperature environment that is warming rapidly due to global change. Little is known, however, on the temperature sensitivity of growth of microbial communities in Antarctic soils and in the surrounding oceanic waters. This is the first study that directly compares temperature adaptation of adjacent marine and terrestrial bacteria in a polar environment. The bacterial communities in the ocean were adapted to lower temperatures than those from nearby soil, with cardinal temperatures for growth in the ocean being the lowest so far reported for microbial communities. This was reflected in lower minimum (T_min) and optimum temperatures (T_opt) for growth in water (?17 and +20°C, respectively) than in soil (?11 and +27°C), with lower sensitivity to changes in temperature (Q₁₀; 0–10°C interval) in Antarctic water (2.7) than in soil (3.9). This is likely due to the more stable low temperature conditions of Antarctic waters than soils, and the fact that maximum in situ temperatures in water are lower than in soils, at least in summer. Importantly, the thermally stable environment of Antarctic marine water makes it feasible to create a single temperature response curve for bacterial communities. This would thus allow for calculations of temperature‐corrected growth rates, and thereby quantifying the influence of factors other than temperature on observed growth rates, as well as predicting the effects of future temperature increases on Antarctic marine bacteria.     

Measurement and mathematical modelling of competition between fast- and slow-growing ordinary heterotrophic organisms in low and high substrate-loaded systems

Ordinary heterotrophic organisms (OHO) of an activated sludge wastewater treatment system showed an atypical growth behaviour when they are inoculated to batch aerobic growth tests with a high substrate-loaded condition. For example, the OHO maximum specific growth rates on readily biodegradable substrates (μ _H) increased with a high ratio of substrate concentration to OHO active biomass concentration (So/Xo), although they were assumed to be constant in a conventional microbial growth kinetic model with a single OHO population group. We, therefore, set a hypothesis in that the change of OHO maximum specific growth rates in the batch test condition is caused by turnover of fast-growing OHO population against slow-growing OHO population. And, a competitive microbial growth kinetic model of the fast- and slow-growing OHO population groups was developed and validated with model-data fitting analysis for the batch test results. The competitive microbial growth kinetic model of process selection, rather than that of kinetic selection, was capable of simulating microbial growth kinetics in high substrate-loaded dynamic systems (i.e., batch tests) and low substrate-loaded steady-state systems (i.e., continuously operated wastewater treatment systems), better than the conventional non-competitive growth kinetic model.     

Birth, growth and death as structuring operators in bacterial population dynamics

A new model is presented that describes microbial population dynamics that emerge from complex interactions among birth, growth and death as oriented, discrete events. Specifically, birth and death act as structuring operators for individual organisms within the population, which become synchronised as age clusters (called cell generations that are structured in age classes) that are born at the same time and die in concert; a pattern very consistent with recent experimental data that show bacterial group death correlates with temporal population dynamics in chemostats operating at carrying capacity. Although the model only assumes “natural death” (i.e., no death from predation or antimicrobial exposure), it indicates that short-term non-linear dynamic behaviour can exist in a bacterial population growing under longer term pseudo-steady-state conditions (a confined dynamic equilibrium). After summarizing traditional assumptions about bacterial aging, simulations of batch, continuous-flow, and bioreactors with recycle are used to show how population dynamics vary as function of hydraulic retention time, microbial kinetics, substrate level, and other factors that cause differential changes in the distribution of living and dead cells within the system. In summary, we show that population structures induced by birth and death (as discrete and delayed events) intrinsically create a non-linear dynamic system, implying that a true steady state can never exist in growing bacterial populations. This conclusion is discussed within the context of process stability in biotechnology.     

Recovery of exponentially growing cultures of Klebsiella pneumoniae NCIB 418 after heat shocks

Exponentially growing cultures of Klebsiella pneumoniae were subjected to heat shocks in the superoptimal and supermaximal temperature ranges for growth on glucose in a defined mineral salts medium. Transitory changes in the specific growth rate constant during recovery were evident. The response was heat shock temperature and exposure time dependent. Cell viability determinations, based on colony counts, indicated complete recovery from heat treatments at superoptimal temperatures. In contrast, at supermaximal temperatures, discrepancies in colony counts on different agars were observed. The kinetic response of the specific growth rate constant after a heat shock at supermaximal temperatures is explained by segregation within the bacterial population     

Temperature-dependent kinetic model for nitrogen-limited wine fermentations

A physical and mathematical model for wine fermentation kinetics was adapted to include the influence of temperature, perhaps the most critical factor influencing fermentation kinetics. The model was based on flask-scale white wine fermentations at different temperatures (11 to 35 degrees C) and different initial concentrations of sugar (265 to 300 g/liter) and nitrogen (70 to 350 mg N/liter). The results show that fermentation temperature and inadequate levels of nitrogen will cause stuck or sluggish fermentations. Model parameters representing cell growth rate, sugar utilization rate, and the inactivation rate of cells in the presence of ethanol are highly temperature dependent. All other variables (yield coefficient of cell mass to utilized nitrogen, yield coefficient of ethanol to utilized sugar, Monod constant for nitrogen-limited growth, and Michaelis-Menten-type constant for sugar transport) were determined to vary insignificantly with temperature. The resulting mathematical model accurately predicts the observed wine fermentation kinetics with respect to different temperatures and different initial conditions, including data from fermentations not used for model development. This is the first wine fermentation model that accurately predicts a transition from sluggish to normal to stuck fermentations as temperature increases from 11 to 35 degrees C. Furthermore, this comprehensive model provides insight into combined effects of time, temperature, and ethanol concentration on yeast (Saccharomyces cerevisiae) activity and physiology.     

A stochastic formulation of the gompertzian growth model for in vitro bactericidal kinetics: parameter estimation and extinction probability

Time-kill curves have frequently been employed to study the antimicrobial effects of antibiotics. The relevance of pharmacodynamic modeling to these investigations has been emphasized in many studies of bactericidal kinetics. Stochastic models are needed that take into account the randomness of the mechanisms of both bacterial growth and bacteria-drug interactions. However, most of the models currently used to describe antibiotic activity against microorganisms are deterministic. In this paper we examine a stochastic differential equation representing a stochastic version of a pharmacodynamic model of bacterial growth undergoing random fluctuations, and derive its solution, mean value and covariance structure. An explicit likelihood function is obtained both when the process is observed continuously over a period of time and when data is sampled at time points, as is the custom in these experimental conditions. Some asymptotic properties of the maximum likelihood estimators for the model parameters are discussed. The model is applied to analyze in vitro time-kill data and to estimate model parameters; the probability of the bacterial population size dropping below some critical threshold is also evaluated. Finally, the relationship between bacterial extinction probability and the pharmacodynamic parameters estimated is discussed.     

Temperature dependence of inorganic nitrogen uptake: reduced affinity for nitrate at suboptimal temperatures in both algae and bacteria.

Nitrate utilization and ammonium utilization were studied by using three algal isolates, six bacterial isolates, and a range of temperatures in chemostat and batch cultures. We quantified affinities for both substrates by determining specific affinities (specific affinity = maximum growth rate/half-saturation constant) based on estimates of kinetic parameters obtained from chemostat experiments. At suboptimal temperatures, the residual concentrations of nitrate in batch cultures and the steady-state concentrations of nitrate in chemostat cultures both increased. The specific affinity for nitrate was strongly dependent on temperature (Q10 approximately 3, where Q10 is the proportional change with a 10 degrees C temperature increase) and consistently decreased at temperatures below the optimum temperature. In contrast, the steady-state concentrations of ammonium remained relatively constant over the same temperature range, and the specific affinity for ammonium exhibited no clear temperature dependence. This is the first time that a consistent effect of low temperature on affinity for nitrate has been identified for psychrophilic, mesophilic, and thermophilic bacteria and algae. The different responses of nitrate uptake and ammonium uptake to temperature imply that there is increasing dependence on ammonium as an inorganic nitrogen source at low temperatures.     

Temperature-Dependent Kinetic Model for Nitrogen-Limited Wine Fermentations

A physical and mathematical model for wine fermentation kinetics was adapted to include the influence of temperature, perhaps the most critical factor influencing fermentation kinetics. The model was based on flask-scale white wine fermentations at different temperatures (11 to 35°C) and different initial concentrations of sugar (265 to 300 g/liter) and nitrogen (70 to 350 mg N/liter). The results show that fermentation temperature and inadequate levels of nitrogen will cause stuck or sluggish fermentations. Model parameters representing cell growth rate, sugar utilization rate, and the inactivation rate of cells in the presence of ethanol are highly temperature dependent. All other variables (yield coefficient of cell mass to utilized nitrogen, yield coefficient of ethanol to utilized sugar, Monod constant for nitrogen-limited growth, and Michaelis-Menten-type constant for sugar transport) were determined to vary insignificantly with temperature. The resulting mathematical model accurately predicts the observed wine fermentation kinetics with respect to different temperatures and different initial conditions, including data from fermentations not used for model development. This is the first wine fermentation model that accurately predicts a transition from sluggish to normal to stuck fermentations as temperature increases from 11 to 35°C. Furthermore, this comprehensive model provides insight into combined effects of time, temperature, and ethanol concentration on yeast (Saccharomyces cerevisiae) activity and physiology.