When is simple good enough: a comparison of the Gompertz,Baranyi, and three-phase linear models for fitting bacterial growth curves

The use of primary mathematical models with curve fitting software is dramatically changing quantitative food microbiology. The two most widely used primary growth models are the Baranyi and Gompertz models. A three-phase linear model was developed to determine how well growth curves could be described using a simpler model. The model divides bacterial growth curves into three phases: the lag and stationary phases where the specific growth rate is zero (gm=0), and the exponential phase where the logarithm of the bacterial population increases linearly with time (gm=constant). The model has four parameters: N_o(Log₈of initial population density), NMAX(Log₈of final population density), tLAG(time when lag phase ends), and tMAX(time when exponential phase ends). A comparison of the linear model was made against the Baranyi and Gompertz models, using established growth data forEscherichia coli0157:H7. The growth curves predicted by the three models showed good agreement. The linear model was more ‘robust' than the others, especially when experimental data were minimal. The physiological assumptions underlying the linear model are discussed, with particular emphasis on assuring that the model is consistent with bacterial behavior both as individual cells and as populations. It is proposed that the transitional behavior of bacteria at the end of the lag phase can be explained on the basis of biological variability.     

Modeling the inactivation of Salmonella typhimurium by dense phase carbon dioxide in carrot juice

The inactivation of Salmonella typhimurium inoculated into acidified carrot juice subjected to dense phase carbon dioxide (DPCD) was investigated. The pressures in the study were 10, 20 and 30 MPa, the temperatures were 32, 37 and 42 °C, and the treatment time was 5–90 min. The inactivation effect of DPCD was enhanced by increasing pressure and temperature. The sigmoid inactivation curves were characterized with the lag phase, exponential inactivation phase, and resistant phase. The inactivation curves were fitted to the modified Gompertz equation and the modified Logistic equation, the modified Gompertz equation was superior since its lowest residual sum of squares (RSS) was lower although there was no significant difference of goodness-of-fit between both models as indicated by F-test. The λ (the duration of the lag phase) and t_4-D (the time necessary to achieve 4-log cycles reduction) decreased with increasing pressure or temperature. The k_dm (the maximum specific value of the inactivation rate, min⁻¹) increased with increasing temperatures, and decreased with increasing pressures. The activation energy (Ea) and the activation volume (Va) necessary for inactivating S. typhimurium by DPCD were 19.06–29.39 kJ mol⁻¹ and 18.89–58.27 cm³ mol⁻¹.     

A new mechanistic growth model for simultaneous determination of lag phase duration and exponential growth rate and a new Bĕlehdrádek-type model for evaluating the effect of temperature on growth rate

A new mechanistic growth model was developed to describe microbial growth under isothermal conditions. The new mathematical model was derived from the basic observation of bacterial growth that may include lag, exponential, and stationary phases. With this model, the lag phase duration and exponential growth rate of a growth curve were simultaneously determined by nonlinear regression. The new model was validated using Listeria monocytogenes and Escherichia coli O157:H7 in broth or meat. Statistical results suggested that both bias factor (B_f) and accuracy factor (A_f) of the new model were very close to 1.0. A new B?lehdrádek-type rate model and the Ratkowsky square-root model were used to describe the temperature dependence of bacterial growth rate. It was observed that the maximum and minimum temperatures were more accurately estimated by a new B?lehdrádek-type rate model. Further, the inverse of square-roots of lag phases was found proportional to temperature, making it possible to estimate the lag phase duration from the growth temperature.     

Description of growth of Clostridium perfringens in cooked beef with multiple linear models

The traditional linear model used in food microbiology employs three linear segments to describe the process of food spoilage and categorize a growth curve into three phases — lag, exponential, and stationary. The linear model is accurate only within certain portions of each phase of a growth process, and can underestimate or overestimate the transitional phases. While sigmoid functions (such as the Gompertz and logistic equations) can be used to fit the experimental growth data more accurately, they fail to indicate the physiological state of bacterial growth. The objective of this paper was to develop a new methodology to describe and categorize accurately the bacterial growth as a process using Clostridium perfringens as a test organism. This methodology utilized five linear segments represented by five linear models to categorize a bacterial growth process into lag, first transitional, exponential, second transitional, and stationary phases. Growth curves described in this paper using multiple linear models were more accurate than the traditional three-segment linear models, and were statistically equivalent to the Gompertz models. With the growth rates of transitional phases set to 1/3 of the exponential phase, the durations of the lag, first transitional, exponential, and second transitional phases in a growth curve described by the new method were correlated linearly. Since this linear relationship was independent of temperature, a complete five-segment growth curve could be generated from the maximum growth rate and a known duration of the first four growth phases. Moreover, the lag phase duration defined by the new method was a linear function of the traditional lag phase duration calculated from the Gompertz equation. With this relationship, the two traditional parameters (lag phase and maximum growth rate) used in a three-segment linear model can be used to generate a more accurate five-segment linear growth curve without involving complicated mathematical calculations.     

The influence of headspace and dissolved oxygen level on growth and haemolytic BL enterotoxin production of a psychrotolerant Bacillus weihenstephanensis isolate on potato based ready-to-eat food products

The major objective of this study was to determine the influence of the initial headspace and dissolved O₂ level and vacuum packaging on growth and diarrhoeal enterotoxin production by Bacillus weihenstephanensis on potato based ready-to-eat food products. In general, the lower the initial headspace or dissolved O₂ level the slower the maximum growth rate (μ_max, log₁₀ CFU g⁻¹ d⁻¹), the longer the lag phase duration (λ, d) and the smaller the maximum population density (N_max, log₁₀ CFU g⁻¹) became. The slowest μ_max, the longest λ and the smallest N_max were generally found for growth under vacuum packaging. This implies shorter shelf-lives will occur at higher initial headspace or dissolved O₂ levels as the growth of B. weihenstephanensis to the infective dose of 10⁵ CFU g⁻¹ in such atmospheres takes a shorter time. Significant consumption of dissolved O₂ only occurred when growth shifted from the lag to the exponential phase and growth generally transitioned from the exponential to the stationary phase when the dissolved O₂ levels fell below ca. 75 ppb. Diarrhoeal enterotoxin production (determined via detection of the L2 component of haemolytic BL) was similar for growth under initial headspace O₂ levels of 1-20.9%, and was only reduced when growth took place under vacuum packaging. The reduction in L2 production when growth took place under vacuum was most probably related to the low final cell densities observed under this condition. Both growth and L2 production were inhibited over a 32-day incubation period at 7 °C by 40% CO₂ irrespective of the headspace or dissolved O₂ levels. The results illustrate the importance of residual O₂ and CO₂ on the shelf-stability and safety of modified atmosphere packaged potato based ready-to-eat food products with regards to B. weihenstephanensis.     

The effect of a competitive microflora,pH and temperature on the growth kinetics of Escherichia coli O157:H7

Growth curves were generated for Escherichia coli O157:H7 in brain–heart infusion broth incubated at 37 or 15°C in the presence of individual and combinations of competing microflora. Broths were inoculated withE. coli O157:H7 (log₁₀3·00 cfu ml⁻¹) and competitors (log₁₀4·00 cfu ml⁻¹) and the initial pH of the broth was either neutral (7·0) or adjusted to 5·8 and then sequentially reduced to 4·8 over 10 h to simulate fermentation conditions. Growth curves were also generated for the competitors in these cultures, including Pseudomonas fragi, Hafnia alvei, Pediococcus acidilactici (pepperoni starter culture) and Brochothrix thermosphacta . Gompertz equations were fitted to the data and growth kinetics including lag phase duration, exponential growth rates and maximum population densities (MPD) calculated. In pure culture, the growth parameters for E. coli O157:H7 in neutral pH broths were significantly different from those recorded in simulated fermentation broths (P<0·05). The presence of competitors in the broth also had a significant effect on the growth kinetics of the pathogen. H. alvei significantly inhibited the growth (lag phase, growth rate and MPD) of E. coli O157:H7 at 37°C, neutral pH and outgrew the pathogen under these conditions. In neutral pH cultures, two other competitors, B. thermosphacta and P. acidilactici also inhibited the lag phase of the pathogen but had no effect on the other growth parameters. In simulated fermentation broths, the growth rate of E. coli O157:H7 was consistently slower and the MPD lower in the presence of a competitive microflora than when grown individually. At 15°C, only one competitor, P. fragi significantly inhibited the lag phase of the pathogen. The implications of these findings for food safety are discussed.     

Prediction of time to growth of Listeria monocytogenes using Monte Carlo simulation or regression analysis,influenced by sublethal heat and recovery conditions

Stochastic models, including the variability in extent and probability of microbial growth, are useful for estimating the risk of foodborne illness (i.e. Nauta, 2000). Risk assessment typically has to embrace all sources of variability. In this paper, a stochastic approach to evaluate growth of heat damaged Listeria monocytogenes cells influenced by different stresses (pH and presence of eugenol) was performed, using an individual-based approach of growth through OD measurements. Both the lag phase duration and the “work to be done” (h₀ parameter) were derived from the growth curves obtained. From results obtained histograms of the lag phase were generated and distributions were fitted. Histograms showed a shift to longer lag phases and an increase in variability with high stress levels. Using the distributions fitted, predictions of time to unacceptable growth (10² cfu/g) of L. monocytogenes were established by Monte Carlo simulation and they were compared with results from statistical methods. It was evidenced that both methods (Monte Carlo and regression analysis) gave a good indication of the probability of a certain level of growth other than the average. Tornado plots were obtained to establish a sensitivity analysis of the influence of the conditions tested (heat, pH, eugenol) applied to the microorganism and their combinations.     

Probability of growth and toxin production by nonproteolytic Clostridium botulinum in rockfish fillets stored under modified atmospheres

Whether toxin production by Clostridium botulinum precedes or follows spoilage of fish stored under modified atmospheres (MA), remains unclear. In this factorial design study we inoculated a pool of nonproteolytic C. botulinum spores (5 type E, 4 type B, and 4 type F strains) at 6 levels (10⁴ to 10⁻¹) between two rockfish fillets and then incubated the fillets at 4, 8, 12 and 30°C under vacuum, 100% CO₂ and 70% CO₂+30% air for 21 days. The probability of toxigenesis by one spore was significantly affected (P<0.005) by temperature (T) and storage time (S_t), and not (P>0.1) by MA, MA×T or MA×S_t. At the 10° spore/sample level, the earliest time to detect toxin production at 4,8,12 and 30°C under all MAs was >21, 15–21, 6–9 and 2 days, respectively. No toxin production was detected at 4°C. Only type B toxin was present in the toxic samples. At 30°C storage, spoilage of fillets followed toxigenesis. Using linear and logistic regression models, equations were derived that could estimate the lag phase and predict the probability of one spore initiating growth under a particular storage condition.     

The effect of abrupt osmotic shifts on the lag phase duration of physiologically distinct populations of Salmonella typhimurium

Relative lag time (RLT), i.e. lag time divided by generation time, was used to characterise the lag phase response of exponential and stationary phase Salmonella typhimurium subjected to NaCl-mediated hyperosmotic shifts. Abrupt hyperosmotic shifts induced lag phases. The RLT, however, varied with the physiological history of the inoculum and the magnitude of the shift. Turbidimetric data showed that exponential phase cells had larger RLTs (up to approximately 8 units) than stationary phase cells (up to 2-4 units). Inocula containing exponential and stationary phase cells mixed in known proportions gave intermediate results. For viable count data, there was little difference in RLT between exponential and stationary phase cells. The RLT response determined turbidimetrically was reproducible for exponential phase cells, but less so for stationary phase cells. It is suggested that there may be a lower limit for resolution of RLT, in the range 0-2 units, and that this may account for the lack of reproducibility in RLTs of stationary phase cells. It is hypothesised that stationary phase cells have enhanced resistance to osmotic stress and are able to exploit new growth environments at low a(w) more rapidly than exponential phase cells, resulting in shorter lag phases. However, the data indicate that turbidimetry may not accurately describe the lag phase response of exponential phase cells subjected to large osmotic shifts. Viable count data is required to investigate this hypothesis further.     

An inhibitory synergistic effect of a nisin–monolaurin combination on Bacillus sp. vegetative cells in milk

The inhibitory activities of nisin and monolaurin, used alone or in combination, were investigated against four Bacillus species vegetative cells in milk at 37°C for 5 days. In the absence of inhibitors, the four strains grew and sporulated at the end of the exponential growth step and throughout the stationary phase. Nisin (100 IU ml⁻¹) induced an immediate reduction in the population level but transient because regrowth appeared and was strain-dependent; cell concentrations reached the control culture level, e.g. 6–7 log₍₁₀₎as well as the spore load (4–5 log₍₁₀₎). On the other hand, monolaurin (250 μ g ml⁻¹) had a durable bacteriostatic effect followed by a regrowth level constantly lower than that of the control culture; sporulation was low (between 13 and 7×10³spl ml⁻¹) and did not occur in the case of B. coagulans. The use of these inhibitors in combination, induced a synergistic bactericidal effect leading to a total inhibition (0 cfu ml⁻¹) until day 5, except in the case of B. cereus where a concentration of 500 cfu ml⁻¹was constant till the end of the experiment; consequently, sporulation was absent.     

Dynamic mechanical thermal analysis of chapati and phulka (Indian unleavened bread)

Glass transition temperature (Tg) and relaxation phenomena (α) of bread, chapati, phulka and phulka containing different anti-staling (A/S) additives such as glycerol, propylene glycol, maltodextrin, and anti-staling enzyme were determined using Dynamic Mechanical Thermal Analysis (DMTA). α-Transition which depends upon order of fall in modulus (FE¹), span (σ) and peak amplitude (A_t) of tan δ transition and Tg was drastically affected between bread and phulka, by incorporation of A/S agents in phulka and during its ageing (staling). During ageing Tg and A_t increased, whereas FE¹ and σ decreased. Phulka containing glycerol, propylene glycol and A/S enzyme gave satisfactory results.     

Modeling the lag phase of Listeria monocytogenes

An estimate of the lag phase duration is an important component for predicting the growth of a bacterium and for creating process models and risk assessments. Most current research and data for predictive modeling programs initiated growth studies with cells grown to the stationary phase in a favorable pH, nutrient and temperature environment. In this work, Listeria monocytogenes Scott A cells were grown in brain heart infusion (BHI) broth at different temperatures from 4 to 37 degrees C to the exponential growth or stationary phases. Additional cells were suspended in a dilute broth, desiccated or frozen. These cells were then transferred to BHI broth at various temperatures from 4 to 37 degrees C and the lag phase durations were determined by enumerating cells at appropriate time intervals. Long lag phases were observed for cells initially grown at high temperatures and transferred to low temperatures. In general, exponential growth cells had the shortest lag phases, stationary phase and starved cells had longer, frozen cells had slightly longer and desiccated cells had the longest lag phases. These data were from immediate temperature transitions. When a computer-controlled water bath linearly changed the temperature from 37 to 5 degrees C over a 3.0- or 6.0-h period, the cells had short lags and grew continuously with declining growth rates. Transitions of 0.75 or 1.0 h had 20-h lag phases, essentially that of immediate transitions. When the transition was 1.5 h, an intermediate pattern of less than 1 log of growth followed by no additional growth for 20 h occurred.     

Application of essential oils in maize grain: impact on Aspergillus section Flavi growth parameters and aflatoxin accumulation

The antifungal activity of Pimpinella anisum L. (anise), Pëumus boldus Mol (boldus), Hedeoma multiflora Benth (mountain thyme), Syzygium aromaticum L. (clove), and Lippia turbinate var. integrifolia (griseb) (poleo) essential oils (EOs) against Aspergillus section Flavi was evaluated in sterile maize grain under different water activity (a_w) condition (0.982, 0.955, and 0.90). The effect of EOs added to maize grains on growth rate, lag phase, and aflatoxin B₁ (AFB₁) accumulation of Aspergillus section Flavi were evaluated at different water activity conditions. The five EOs analyzed have been shown to influence lag phase and growth rate. Their efficacy depended mainly on the essential oil concentrations and substrate water activity conditions. All EOs showed significant impact on AFB₁ accumulation. This effect was closely dependent on the water activity, concentration, and incubation periods. Important reduction of AFB₁ accumulation was observed in the majority of EO treatments at 11 days of incubation. Boldus, poleo, and mountain thyme EO completely inhibited AFB₁ at 2000 and 3000 μg g⁻¹. Inhibition of AFB₁ accumulation was also observed when aflatoxigenic isolates grew with different concentration of EOs during 35 days.     

Response of mixed-age cultures of phosphine-resistant and susceptible strains of lesser grain borer, Rhyzopertha dominica, to phosphine at a range of concentrations and exposure periods

Mixed-age cultures, containing all life stages, of a highly resistant strain (Strong-R) of lesser grain borer, Rhyzopertha dominica (F.), were exposed to a series of fixed concentrations of phosphine at a range of exposure periods at 25 °C. A susceptible strain and a less-resistant strain (Weak-R) were also tested. The aim was to characterise the resistant strain and determine if it could be controlled with phosphine. Times to population extinction (TPE) were recorded and lethal time (LT_99.9) values calculated. The relationship between exposure period t and phosphine concentration C for the resistant strains were: for Strong-R LT_99.9C^0.5457t=3.852 and TPE C^0.6105t=4.0404 and for Weak-R LT_99.9C^0.3553t=3.6521 and TPE C^0.4507t=3.4833. The results were used to define a range of minimum exposure period ×concentration protocols for control of the Strong-R populations. For example, at 1.0, 0.3, and 0.2 mg l⁻¹ complete control can be expected in 5, 10 and 14 days, respectively. This information will be used to recommend phosphine rates for field trial and eventual registration.     

Numerical and Statistical Methodology to Analyze Microbial Spoilage of Refrigerated Solid Foods Exposed to Temperature Abuse

Numerical and statistical procedures based on pseudo-zero for the lag and first order reaction kinetics for the exponential growth phase were developed to analyze non-isothermal microbial spoilage. Arrhenius model parameters and their accuracy were estimated for a mixture of Pseudomonas fluorescens, Staphylococcus aureus andAchromobacter lwoffi growing in a seafood model. Linear regressions used with isothermal experiments generated initial values for nonlinear estimations of the frequency (K₀) and activiation energy (E_a) constants. An optimization technique was used to minimize the square difference between experimental and estimated values while parameter accuracy was assessed using a bootstrap method. E_a and In(K₀) were 109±3.4 and 48.3±1.5 kJ/mole for the exponential, and 152±4.0 and 64.4±1.7 kJ/mole for the lag phase, respectively. The Mann-Whitney-Wilcoxon rank sum test showed no significant differences between parameters generated by two different temperature profiles (5% significance level).     

Effects of ultrasound on the fermentation profile and metabolic activity of lactic acid bacteria in buffalo's (Bubalus bubalis) milk

In this study, the ultrasound was applied at the lag phase and logarithmic phase of the growth of Streptococcus thermophilus subsp. Salivarius (6.76 ± 0.15 log cfu/mL) and Lactobacillus bulgaricus (6.36 ± 0.22 log cfu/mL) in buffalo's milk. The results revealed that S. thermophilus entered into the logarithmic phase after 45 min of incubation. Ultrasound application during lag phase (1685 J mL⁻¹) and logarithmic phase (561.6 J mL⁻¹) reduced the fermentation time by 32 min (12.5%) and 40 min (15.7%), respectively. Ultrasound treatment (1684.8 J mL^-1) during the lag phase resulted in fractures (approximately 0.1–0.2 μm) on the cell membrane of S. thermophilus. Application of ultrasound for a 2–6 min period enhanced (P < 0.05) the microbial growth while increasing (P < 0.05) the β-galactosidase activity. The lactose hydrolysis was increased up to 49.2% compared to that of the control. Ultrasound treatment during the lag phase showed higher (P < 0.05) storage modulus, yield point and consistency index compared to the logarithmic phase. It can be concluded that ultrasound has huge potential to improve the fermentation rate of buffalo's milk.Industrial relevanceConventional food fermentation is one of the most time and resources consuming processing steps in the food industry. In this work, it was shown that the appropriate application of ultrasound during different stages of microbial growth reduced the fermentation time of buffalo milk by 13% to 16% by enhancing the metabolic activities of lactic acid bacteria. Therefore, ultrasound has great prospects in the food processing industry to improve process efficiency. Scaling-up of laboratory applications of US in to industrial scale through designing new power transducers with high capacity and greater acoustic filtration is necessary to enhance its potential to be used at large scales.     

Modelling the effect of a temperature shift on the lag phase duration of Listeria monocytogenes

The aim of this work is to study and model the effect of a temperature shift on h₀, the product of the growth rate by the lag phase duration (μλ). Our work is based on the data of Whiting and Bagi [Int. J. Food Microbiol. 73 (2002) 291], who studied the influence of both the pre-incubation temperature (T_prior) and the growth temperature (T_growth) on λ values of Listeria monocytogenes. We introduce a new model to describe the evolution of the parameter h₀ as a function of T_prior and T_growth, and compare it to Whiting and Bagi's published polynomial model that describes the influence of T_prior and T_growth on λ independently of μ. For exponential as well as stationary phase cells, h₀ increases almost linearly with the magnitude of the temperature shift. A simple linear model of h₀ turns out to be more suitable to predict λ values than a polynomial model of λ.     

Estimation of the survival curve of Listeria monocytogenes during non-isothermal heat treatments

Published survival curves of Listeria monocytogenes under several constant temperatures in the range of 50–65°C could be described by the model Log₁₀[S(t)]=−b(T)t^n(T), where S(t) is the momentary survival ratio, and b(T) and n(T) coefficients whose temperature dependence was expressed by empirical models. When the temperature history T(t) is also expressed algebraically, b(T) and n(T) are transformed into time dependent terms, b[T(t)] and n[T(t)] respectively. If there is no growth and damage repair during the heating process, and the momentary inactivation rate only depends on the momentary temperature and survival ratio, then the solution of the differential equation dLog₁₀[S(t)]/dt=−b[T(t)]*n[T(t)]*{−Log₁₀[S(t)]/b[T(t)]}^∧{(n[T(t)]−1)/n[T(t)]} provides the survival curve under the specified non-isothermal conditions. The validity of this model is demonstrated by the agreement of its predictions, calculated numerically using Mathematica®, to reported survival data of Listeria during heating at a constant and varying rates. Unlike in the traditional calculation methods of microbial survival, the one employed here does not require that microbial mortality be a process following a first or any other order kinetics model.     

Physiological parameters of Bacillus cereus marking the end of acid-induced lag phases

During lag phases microbial cells adapt to their environment and prepare to proliferate. Physiological parameters of B. cereus cells upon exposure to near-growth-boundary acid stress were investigated and markers for the transition between lag phase and growth were identified using fluorescent probes combined with flow cytometry. Determination of cell counts and optical density revealed lag phases of 1 h, 2 h and 5 h, in cultures shifted to pH 7, pH 5.3 (set with lactic acid) and pH 4.9 (set with sulfuric acid), respectively. The obtained lag phases fitted the trends in ATP levels, which were constant during the lag phase and increased after the onset of growth. Both the percentage of PI-stained cells and cells with a significant membrane potential decreased during the lag phase. This points to repair of membrane damage and the loss of membrane potential. However, both trends extended in the growth phase, thus not suitable to mark the onset of growth. The activity of the electron transfer chain and esterases did allow for assessment of transition between lag and growth phase. These activities were generally low during the lag phase and increased after the onset of growth. Our results show that, independent of the duration of the lag phase, for different conditions the same physiological trends could be observed. The change in signal of selected probes can be used as a marker for transition from lag phase to the growth phase and may aid in identification of novel targets interfering with bacterial exit from lag phase.