International Journal of Food Microbiology, Volume 57, Issue 3 , 15 June 2000, Pages 169-181

A model describing the effect of  temperature history  on lag time  for Listeria monocytogenes

Jean-Christophe Augustin^a, Laurent Rosso^b and Vincent Carlier<sup>a

a</sup> Service d’Hygiène et Industrie des Denrées Alimentaires d’Origine Animale, Ecole Nationale Vétérinaire d’Alfort, 7 avenue du Général de Gaulle, F-94704 Maisons-Alfort Cedex, France
^b Pôle de Compétence en Sécurité des Aliments, Groupe Danone, Z.I. Le Teinchurier, F-19100 Brive-la-Gaillarde, France

Received 13 November 1999; revised 25 January 2000; accepted 8 February 2000. Available online 23 May 2000.

ABSTRACT

where x₀ is the initial bacterial concentration (cfu ml⁻¹) estimated from the average of two initial enumerations, x_max is the maximum bacterial concentration (cfu ml⁻¹), x(t) is the bacterial concentration (cfu ml⁻¹) at time t (h), _max is the maximum specific growth rate (h⁻¹), and lag is the lag time (h).

The lag time was estimated as being the average of two values obtained with two bacterial concentrations in the log phase of growth (Fig. 1). _max was estimated by applying an exponential model with delay on optical densities in the range 0 to 0.1 and by correcting the growth rate obtained by a calibration factor.

For incubation on TSYE agar, cells present on the surface of the agar were collected and suspended in 5 ml of bacto-peptone (Difco Laboratories, Detroit, MI) plus 0.85% sodium chloride (Prolabo, Paris, France) (PS) and appropriate dilutions in PS were plated on TSYE agar plates with a model DS spiral plater (Interscience, St Nom la Bretèche, France). TSYE agar plates were incubated at 37°C for 24 h before enumeration. Growth curves were fitted using the logistic growth model with delay, i.e. with a breakpoint at the transition between the lag and the exponential phase (Kono; Baranyi and Rosso):

2.4. Development of the model describing the relationship between the pre-incubation conditions and the subsequent lag phase duration

The model was built for a pre-incubation in TSYE broth at 4.4, 6.3, 7.8, 10.1, 11.9, 14.0, 15.5, 24.0, 30.9, and 35.9°C and an incubation in TSYE broth at 6°C with the strain Scott A. Experiments at each pre-incubation temperature were replicated between two and four times.

As previously reported (Buchanan; Hudson and Dufrenne) the growth rate was not influenced by pre-incubation conditions, the model was then developed for the product lag×_max, called .

Whatever the temperature of pre-incubation used, a similar evolution of with the duration of the pre-incubation was observed: decreased during the lag phase of growth of the inoculum to approximately 0, stayed at this minimal value during the log phase and sometimes the beginning of the stationary phase and gradually increased during the stationary or death phase.

The growth (increase in cell number) was assumed to be controlled by a factor which must attain a critical concentration before cell division occurs and this factor was assumed to be correlated to the cell biomass (Albertson; Cooper; Baranyi; Baranyi; Hills and Hills). The biomass was assumed to increase exponentially during the lag phase with a rate equal to _max (Koch; Koch; Hills; Baranyi and Hills). Growth is then delayed during the lag phase until the biomass has reached its critical value. If during the lag phase, cells are abruptly placed in other incubation conditions, the rate of biomass changes instantaneously to the new growth rate (Baranyi et al., 1995) and a new lag time can be defined as the time necessary to increase from the initial biomass to the critical one. The product is then described by the following equation:

where _i (h⁻¹) is the maximum specific growth rate of the inoculum at the temperature T_i (°C), lag_i (h) is the lag time of the inoculum at the temperature T_i, and t_i (h) is the duration of the pre-incubation at the temperature T_i.

During the exponential growth phase (t_i>lag_i) of the inoculum the product is equal to 0 as the factor controlling the cell division is above its critical value. This phenomenon was first reported by Penfold (1914).

When the inoculum enters into the stationary phase, the biomass decreases (Cooper; Matin; Siegele; Kolter; Hills; Hills and Koch), cell division ceases and the product will then increase. When the bacterial cells are starved, increases to infinity (Albertson et al., 1990). The beginning of the increase of was then assumed to be linked to the entry of the inoculum into the stationary phase and the rate of increase of was assumed to be linked to the maximum specific growth rate of the inoculum. A square root relationship was found to be suitable to describe the increase of with the pre-incubation duration:

where f and g are secondary models describing the evolution of k and t_s.

The entry into the stationary phase was arbitrarily defined as the intercept between the extrapolated straight line, slope of which is the growth rate, at the lag time and the natural logarithm of the maximum bacterial concentration horizontal asymptote.

As t_s can theoretically be lower than lag_i, the global model describing the evolution of the product with the pre-incubation duration, t_i, at the temperature T_i is:

with k=f(_i) and .

2.5. Model fits

Fits were performed by linear or non-linear regression using the least squares criterion (Box et al., 1978). Estimation of parameters was carried out by minimizing the sum of the squared residuals (SSR) where SSR is defined as follows:

where n is the number of data points.

The minimum SSR values were computed with the and subroutines of MATLAB 5.2 software (The MathWorks Inc., Natick, MA, USA).

2.6. Validation of the model

The model proposed was built with the strain Scott A with a pre-incubation in liquid medium and an incubation in liquid medium at 6°C. The validity of this model was then checked for other strains, other incubation temperatures, and solid pre-incubation and incubation media.

The conditions tested were:

(i) pre-incubation in TSYE broth at 15.5°C and incubation in TSYE broth at 6°C with strains CLIP 22485, CLIP 19884, 27795, and 925318

(ii) pre-incubation in TSYE broth at 19.6 and 30.9°C, and incubation in TSYE broth at, respectively, 7.8 and 15.5°C

(iii) pre-incubation in TSYE broth at 19.6 and 30.9°C, and incubation on TSYE agar at, respectively, 7.8 and 15.5°C

(iv) pre-incubation on TSYE agar at 19.6 and 15.5°C, and incubation in TSYE broth at, respectively, 6 and 7.8°C.

The validity of the proposed model was also checked with published data obtained for L. monocytogenes.

3. RESULTS

3.1. Evolution of during the lag phase of the inoculum

A linear decrease of to 0 was effectively observed during the lag phase of the inoculum (Fig. 2). This reinforced the hypothesis of the existence of a critical factor increasing exponentially during the lag phase.

 

Fig. 2. Evolution of the product lag×_max for regrowth of L. monocytogenes in TSYE broth at 6°C for inoculated cells in the lag phase cultured in TSYE broth at (a) 11.9°C and (b) 15.5°C.

 

3.2. Evolution of during the stationary phase of the inoculum

A linear increase of ² with the duration of pre-incubation was effectively observed for inoculated cells in stationary phase (Fig. 3). The slope of the line, k, increased with the temperature and then with _i. The time at which the increase began, t_s, decreased with temperature and was then assumed linked to the entry into the stationary phase.

 

Fig. 3. Evolution of the square of the product lag×_max for regrowth of L. monocytogenes in TSYE broth at 6°C for inoculated cells in the stationary phase cultured in TSYE broth at (•) 4.4°C, () 6.3°C, and () 10.1°C.

 

Plots of estimated k against _i are shown in Fig. 4. A good linear correlation (linear correlation coefficient of 0.993) between the estimated rates was observed. The constant term of the regression line was not significantly different from 0 so it was neglected and the two rates were assumed proportional.

 

Fig. 4. Evolution of the rate (k) of increase of the product lag×_max for regrowth of L. monocytogenes in TSYE broth at 6°C with the maximum specific growth rate of the inoculum (_i).

 

Plots of estimated t_s against time of entry into the stationary phase are shown in Fig. 5. A linear correlation (coefficient of 0.970) was also observed but the times were not proportional and negative t_s were observed for high pre-incubation temperatures corresponding to rapid entries into the stationary phase. Given the small size of the available data set, large magnitude of the 95% confidence intervals were obtained for the estimated parameters.

 

Fig. 5. Evolution of the start (t_s) of the increase of the product lag×_max for regrowth of L. monocytogenes in TSYE broth at 6°C with the time of entry into the stationary phase of the inoculum (t_station).

 

3.3. Global model

The global model describing the evolution of with the duration t_i of the pre-incubation at the temperature T_i was then:

with k=0.035·_i and

Knowing the growth parameters of the inoculum, theoretical evolutions of with the duration of pre-incubation were built at different temperatures (Fig. 6). Theoretical values were relatively consistent with the observed ones (Fig. 7) and a root mean square error of 0.61 was obtained for the 172 data points.

 

Fig. 6. Evolution of the product lag×_max for regrowth of L. monocytogenes in TSYE broth at 6°C with the duration of the pre-incubation at (a) 4.4°C, (b) 6.3°C, (c) 7.8°C, (d) 10.1°C, (e) 11.9°C, (f) 14.0°C, (g) 15.5°C, (h) 24.0°C, (i) 30.9°C, and (j) 35.9°C.

 

Fig. 7. Plots of theoretical against observed products lag×_max for regrowth of L. monocytogenes in TSYE broth at 6°C.

 

3.4. Validation of the global model

The validity of the model was checked with four other L. monocytogenes strains for a pre-incubation in TSYE broth at 15.5°C and an incubation in TSYE broth at 6°C. The predicted values for were consistent with the observed ones (Fig. 8) and a root mean square error of 0.88 was obtained.

 

Fig. 8. Plots of predicted against observed products lag×_max for regrowth of L. monocytogenes (•) in TSYE broth at 6°C for strains CLIP 22485, CLIP 19884, 27795, and 925318 pre-incubated in TSYE broth at 15.5°C, () in TSYE broth at 7.8 and 15.5°C for pre-incubation in TSYE broth at, respectively, 19.6 and 30.9°C, () on TSYE agar at 7.8 and 15.5°C for pre-incubation in TSYE broth at, respectively, 19.6 and 30.9°C, (+) in TSYE broth at 6 and 7.8°C for pre-incubation on TSYE agar at, respectively, 19.6 and 15.5°C.

 

The validity was checked for incubations in TSYE broth at 7.8 and 15.5°C with pre-incubations in TSYE broth at, respectively, 19.6 and 30.9°C. The predicted values obtained were also consistent with the observed ones (Fig. 8) and a root mean square error of 0.51 was obtained.

When the incubation was done on TSYE agar at 8 and 16°C with pre-incubations in TSYE broth at, respectively, 20 and 30°C, the predicted values for were distant from the observed ones (Fig. 8) and a root mean square error of 1.48 was obtained. The observed increase of was faster than the predicted one (Fig. 9a).

 

Fig. 9. Evolution of the product lag×_max for regrowth of L. monocytogenes (a) on TSYE agar at 7.8°C with the duration of the pre-incubation in TSYE broth at 19.6°C, (b) in TSYE broth at 6°C with the duration of the pre-incubation on TSYE agar at 15.5°C.

 

When the pre-incubation was done on TSYE agar at 15.5 and 19.6°C with incubations in TSYE broth at, respectively, 6 and 7.8°C, the predicted values for were also distant from the observed ones (Fig. 8) and a root mean square error of 1.06 was obtained. As for the incubation on solid medium, the observed increase of was faster than the predicted one (Fig. 9b)

3.5. Validation of the model with published data

For predictions, when the pre-incubation parameters were not specified in the published studies, the following values were arbitrarily taken: x₀=10⁶ cfu ml⁻¹, x_max=10⁹ cfu ml⁻¹, and lag_i and _i set at the values observed or estimated with the square root model (Ratkowsky and Zwietering) for the strain Scott A in our experiments.

The proposed model is consistent with the result of George and Lund (1992) who observed no lag phase when cells in the log phase at 20°C were transferred to fresh medium.

Membré et al. (1999) observed no lag phase when L. monocytogenes cells were cultured at 4 and 7°C for, respectively, 6 and 4 days and then transferred to fresh medium at 7°C but a lag time of ~31 h was obtained with a pre-incubation at 37°C for 20 h (=1.86 with _max assumed to be 0.06 h⁻¹). The model effectively predicted no lag phase for pre-incubations at 4 and 7°C and a value of 1.26 for was predicted for the pre-incubation at 37°C.

Buchanan and Klawitter (1991) observed an increase of lag phase duration with increasing pre-incubation temperature. The model predicted an evolution of the product similar to those observed but predicted values were under-estimated (Fig. 10). This discrepancy is probably due to the transfer to a different medium which induces a medium-dependent component assumed additive.

 

Fig. 10. Plots of predicted and observed products lag×_max for regrowth of L. monocytogenes at 5°C against the pre-incubation temperature of the inoculum. (•) and () are products lag×_max obtained by Buchanan and Klawitter (1991) in, respectively, aerobic and anaerobic conditions. (+) are predicted products. The durations of the pre-incubations are 144 h at 5°C, 96 h at 10 and 13°C, 48 h at 19°C, 24 h at 28, 37, and 42°C.

 

The same phenomenon was observed with the results obtained by Walker et al. (1990), however the predicted increase of 1.43 by using a pre-incubation at 30°C for 48 h instead of one at 4°C for 6 days was consistent with the average increase observed of 1.69.

4. DISCUSSION

The root mean square error of the model corresponding to the average error (Baranyi and Roberts, 1995) in estimates is 0.61, i.e. estimated values are on average 0.61 away from the observed ones. Thus the goodness of fit of the model seems poor but large measurement errors were observed for estimates. Indeed, the standard error observed with 31 estimated values for a pre-incubation at 30°C for 24 h was 0.46.

The model proposed for the evolution of the product lag×_max is consistent with the hypothesis of an exponential accumulation of a critical substance during the lag phase of which production is instantaneously adjusted to the new environment temperature. This observation supports the usual procedure used to calculate lag time for cultures growing under changing temperatures, which consists of summing lag times for each time/temperature interval (Smith; Li; Rosso and Wijtzes). Although a positive history effect was observed for the lag time of Pseudomonas fragi under temperature transitions between 4°C and 16°C (Fu and Labuza), results obtained in our study are consistent with those of Li and Torres (1993) and Baranyi et al. (1995) who observed no history effect during the lag phase. This observation implies an instantaneous adjustment of the metabolism during the lag phase.

It is usually assumed that the growth rate of exponentially growing cells adapts instantaneously to changes in temperature but lags induced by temperature shifts-down were frequently observed. For exponentially growing cultures of Salmonella typhimurium,Lark and Maaløe (1954) observed no lag phase by reducing the temperature from 37 to 25°C, but observed a lag by reducing the temperature from 37 to 13 or 10°C. This observation was confirmed with Escherichia coli by Ng et al. (1962) and Jones et al. (1987) who observed a lag before growth resumed when the temperature was shifted from 37 to 10°C. Baranyi et al. (1995) observed no induced lag for exponentially growing cultures of Brochothrix thermosphacta for a temperature shift-down from 17–25°C to 5°C but a significant lag was observed for a shift-down to 3°C. The magnitude of this cold shock response is dependent upon the range of the temperature shift (Gounot and Berry). Thus Li and Torres (1993) observed no induced lag with B. thermosphacta for fluctuating temperatures between 4 and 12°C or 2 and 14°C. The range of the temperature shift inducing a cold shock response is not known for L. monocytogenes, this response has been shown for temperature shift-down from 37 to 5°C (Bayles et al., 1996) or from 25 to 4°C ( Phan-Thanh and Gormon, 1995). The proposed model predicted no nil regrowth lag times for exponentially growing cells at high temperature (Fig. 6). Indeed, for rapid entries into the stationary phase, i.e. high pre-incubation temperatures, t_s values are widely negative so the increase of predicted regrowth lag time is observed before the end of the lag phase of the inoculated cells and nil values are not obtained. Nevertheless, as the time of entry in the stationary phase is inoculum-size-dependent, the model can predict nil regrowth lag time for high pre-incubation temperatures when the pre-incubation inoculum (x₀) is very small. A refinement of the model could be to model t_s as a function of both the entry into the stationary phase and the temperature of pre-incubation (or a linked parameter). However, the modelling of this cold shock response seems difficult because even for pre-incubation temperatures of 30.9 and 35.9°C, nil regrowth lag times were sometimes observed (Fig. 6).

The critical substance responsible for the bacterial division does not seem to be correlated to the per cell biomass during the stationary phase. Indeed the per cell biomass decreases before the entry into the stationary phase to attain its minimal value (Siegele; Kolter and Hills) but nil regrowth lag times were observed for cells in stationary growth phase for varying durations at low pre-incubation temperatures. The descriptive term ‘stationary phase’ corresponds effectively to a period of highly heterogeneous cell physiology (Kolter et al., 1993). For prolonged starvation, a gradual increase of the product lag×_max was observed with the starvation duration as it has been previously reported for Vibrio sp. S14 (Albertson et al., 1990). This increase seemed faster for pre-incubation or incubation on a solid medium. For pre-incubation on a solid medium, this observation can be explained by the fact that there is a great heterogeneity in bacterial colonies and cells at the center of the colonies are rapidly in starvation conditions because of the restricted nutrient diffusion ( McKay et al., 1997). Contrary to the pre-incubation, no explanation was available for the phenomenon observed with incubation on a solid medium.

By using this model and the model developed by Bréand et al. (1999) describing the influence of heating on regrowth lag time, it is possible to take into account the influence of the physiological state of inocula in predictive models. These models should be complemented to include the influence of other physical or chemical stress to improve the value of predictive microbiology.

ACKNOWLEDGEMENTS

This work was supported by the Association Vétérinaire d’Hygiène Alimentaire. We would like to thank Cécile Lahellec, Olivier Cerf, and Pierre Pardon who initiated this work, and Tobin Robinson for the helpful and critical reading of the manuscript.

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