Modeling the Lag Time of Listeria monocytogenes from Viable Count Enumeration and Optical Density Data

The following two factors significantly influence estimatesof the maximum specific growth rate (µ_max) and the lag-phaseduration ( ${lambda}$ ): (i) the technique used to monitor bacterial growthand (ii) the model fitted to estimate parameters. In this study,nine strains of Listeria monocytogenes were monitored simultaneouslyby optical density (OD) analysis and by viable count enumeration(VCE) analysis. Four usual growth models were fitted to ourdata, and estimates of growth parameters were compared fromone model to another and from one monitoring technique to another.Our results show that growth parameter estimates depended onthe model used to fit data, whereas there were no systematicvariations in the estimates of µ_max and ${lambda}$ when the estimateswere based on OD data instead of VCE data. By studying the evolutionof OD and VCE simultaneously, we found that while log OD/VCEremained constant for some of our experiments, a visible linearincrease occurred during the lag phase for other experiments.We developed a global model that fits both OD and VCE data.This model enabled us to detect for some of our strains an increasein OD during the lag phase. If not taken into account, thisphenomenon may lead to an underestimate of ${lambda}$ .

INTRODUCTION

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Introduction

The following two methods are commonly used to monitor growthof bacteria: viable count enumeration (VCE) and absorbance measurement.Monitoring bacterial growth by VCE is time-consuming and ratherexpensive, but it remains the method of reference. Methods basedon absorbance measurements constitute a second family of methodsbased on the direct proportionality between the optical density(OD) of a liquid medium and the concentration of bacteria. ODtechniques are rapid, convenient, and inexpensive. However,many drawbacks are inherent to these techniques. The main problemencountered is the limited range of validity since the detectionthreshold typically corresponds to a bacterial concentrationgreater than 10⁶ bacteria/ml (8).

From these two kinds of data, characteristic growth parameters,mainly the lag-phase duration ( ${lambda}$ ) and the maximum specific growthrate (µ_max), can be assessed. The use of mathematicalgrowth models allows accurate and objective estimation of theseparameters. In the field of predictive microbiology, numerousmodels have been developed. Mechanistic models are especiallyinteresting as they provide both a method of estimating ${lambda}$ andµ_max and a means of understanding bacterial growth.

In fact, the following two potential sources of bias influenceestimation of growth parameters: the type of data (OD or VCE)and the model used to fit data. Because of the high detectionthreshold of OD techniques, the initial inoculum must be largeenough to allow reliable measurements, and the question whichhas arisen is whether the estimates of µ_max at high concentrationsare not systematically lower than the actual µ_max becauseof possible end-of-growth inhibition. Nevertheless, this phenomenonseems to have no effect on the estimates of µ_max (9).Hudson and Mott (14) fitted the modified Gompertz equation toPseudomonas fragi growth data sets and obtained significantlylower ${lambda}$ estimates from OD data than from VCE data. Accordingto these authors, the discrepancy between ${lambda}$ values measured byOD and VCE is due to an increase in cell length during the lagphase. This problem was solved by proposing a linear calibrationfunction. With regard to the µ_max, Hudson and Mott (14)found that estimates derived from OD and VCE data were verysimilar, whereas Dalgaard et al. (12) showed that OD-based estimatesof µ_max are systematically lower than VCE-based estimatesand that the discrepancy differs from one model to another.In fact, the accuracy of estimates of µ_max and ${lambda}$ closelydepends on the model chosen to fit data (11). In particular,several models proved to be limited in terms of providing preciseestimates of growth parameters from absorbance data, whereasother models appeared to be quite relevant. Augustin et al.(1) pointed out that reliable estimates of µ_max couldbe obtained by using a calibration factor constant for Listeriamonocytogenes strains. On the other hand, these authors proposedan original method that combines OD and VCE measurements forestimation of ${lambda}$ .

Overall, many authors have emphasized the important variabilityof growth parameter estimates (especially ${lambda}$ estimates) due tothe method used to acquire growth data and to the nature ofthe model used to fit the growth data.

The aim of the present study was to improve our knowledge concerningthe lag phase by acquiring precise OD and VCE growth data simultaneouslyfrom the same bacterial culture. We focused on the first stagesof growth, namely, the lag phase and the beginning of exponentialgrowth. Various models were fitted to our data in order to obtaininformation on the quality of fit and on the accuracy for estimatingthe growth parameters. Particular attention was paid to findingout whether more precise data might help us select an optimalmodel more easily. In addition, a dynamic study of the evolutionof the OD/VCE ratio allowed us to draw attention to an exponentialincrease in OD that happened during the lag phase for one-halfof our strains. If this phenomenon is not taken into accountby modeling, it may lead to underestimation of ${lambda}$ . We proposea new model that fits OD and VCE data globally, estimates asingle value for µ_max and ${lambda}$ , and, if necessary, accountsfor the exponential increase in OD.

MATERIALS AND METHODS

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Materials and Methods

Bacterial strains and growth experiments.
Nine strains of L. monocytogenes were studied. Table 1 providesa brief description of these strains.

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TABLE 1. Description of strains

Stock cultures were maintained at -196°C in brain heartinfusion broth (bioMérieux, Marcy l'Etoile, France) containing10% glycerol. Prior to each experiment, the isolate was grownfor 20 h at 20°C in 15 ml of Trypticase soy broth (bioMérieux).This first bacterial culture was diluted in 230 ml of Trypticasesoy broth in order to obtain an initial bacterial density ofapproximately 10⁷ cells/ml. The initial bacterial density waschosen carefully so that it was greater than the OD detectionthreshold. After inoculation, the strains were grown for 8 hat an incubation temperature of 20°C. The pre- and postincubationconditions were purposely identical in order to reduce sourcesof variability.

Growth was monitored by obtaining turbidimetric measurements(BioPhotometer 6131; Eppendorf) every 15 min. Concurrently,100 µl of culture was removed, diluted, and plated induplicate for VCE.

In addition, in order to ascertain the infrastrain reproducibilityof phenomena, three randomly chosen strains (III 10126, III10111, and IV 644) were monitored twice. Duplicates of thesestrains, which we designated III 10126b, III 10111b, and IV644b, were treated in the same way as the other strains, andsubsequent analyses took these duplicates into account.

Individual fit of models with OD and VCE data.
Four growth models were fitted to our OD and VCE growth data.These models allowed us to describe the lag phase independentof the rest of the growth kinetics (which is not possible whenempirical models, such as the Gompertz and the Logistic models,are used). They also possess biologically interpretable parameters.Explicit equations of these models are described below.

The simplest model is the exponential model. It describes onlythe exponential phase and does not take into account any lagphase:

(1)
where x(t) is the bacterialdensity (in cells per milliliter) at time t (in hours), x₀ isthe initial bacterial density (in cells per milliliter), andµ_max is the exponential growth rate (in hours^-1).

The second model which we fitted is an exponential model withdelay (ED model), which has been described with different namesby various authors (2, 10, 17). Curves obtained from this modelshow an abrupt transition between the lag phase and the exponentialphase:

(2)
The third model is the modelproposed by Baranyi and Roberts (4). Within the framework ofour study we used this model without the inhibition function.This model has three parameters:

(3)
Thefourth equation was proposed by Hills and Wright (13). An identicalequation results from another model proposed by Baranyi (2,3). Like the two previous models, this model has three parameters:

(4)

Global model fit with OD and VCE data.
The model constructed for this study is based on the assumptionthat there is direct proportionality between turbidimetric measurementsobtained by OD (x_OD) and VCE (x_VCE): x_OD(t) = kx_VCE(t).

As OD and VCE growth data were obtained simultaneously fromthe same bacterial cultures we were able to fit a global modelto both types of data. By assuming that the proportional relationshipdescribed above was true, a simple global model, called thepartial model, was defined:

(5)
wheref is a usual growth model function (equations 1 to 4), ${theta}$ is thevector of growth parameters (x₀, ${lambda}$ , and µ_max), and k isthe x_OD(t)/x_VCE(t) ratio.

From the partial model (equation 5), we decided to constructanother more complex model, which we called the full model.In this model we made the assumption that the OD may increaseduring the lag phase. We also made the assumption that biomasskinetics can be split into two distinct phases. The first phaseoccurs during the lag phase and corresponds to an exponentialincrease in the cell biomass of the nondividing cells. The secondphase corresponds to an exponential increase in the biomassdue to the successive cell divisions that happen after a delay ${lambda}$ .

(6)
where µ₁ is the rate ofincrease in the cell OD during the lag phase.

The partial model (equation 5) is nested in the full model (equation6) as it corresponds to the peculiar case in which the rateof increase in the OD during the lag phase (µ₁) is zero.

Fitting procedures and statistical methods.
Stabilization of the variance of the VCE and OD data was doneby using the usual logarithmic transformation. Fits of modelsto the log-transformed data were performed by nonlinear regressionby using the least-squares criterion. Estimates for parameterswere obtained by minimizing the residual sum of squares (RSS):

where N is the number of data points, y_i is theobserved value, and $y$ _i is the fitted value.

Our new model was fitted globally to OD and VCE data. To dothis, in each data set we added a control variable describingthe type of data (1 for OD and 2 for VCE). Nonlinear regressionwas computed with the NonLinearRegress function of Mathematica(Wolfram Research) that uses the Levenberg-Marquardt algorithm.

The performance of the models was evaluated by using a comparisonof root mean square error (RMSE) () between experimental and predicted data. Additional graphic analysisof residuals and Beale's confidence regions (7) were applied.We also studied the precision of parameter estimates in termsof asymptotic marginal confidence intervals.

To decide which is the simplest nested model to fit data adequately,we used an F test (6):

where n is thedata set size, p_f is the number of parameters of the full model,p_p is the number of parameters of the partial model, RSS_f isthe residual sum of squares of the full model fit, and RSS_pis the residual sum of squares of the partial model fit. Theobserved F value must be compared with a theoretical F valuewith ${nu}$ ₁ = p_f - p_p and ${nu}$ ₂ = n - p_f degrees of freedom.

We compared the growth parameters and the RMSE obtained by fittingthe different models to OD and VCE data by carrying out analysisof variance tests ( ${alpha}$ = 5%) for a randomized block design. Finally,we used a paired t test ( ${alpha}$ = 5%) to compare estimates of ${lambda}$ andµ_max obtained from OD and VCE data.

All the statistical tests were performed by using R softwareroutines (version 1.3.1) (15).

RESULTS

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Results

Individual fits of the exponential model and the ED model.
Fits of the exponential model and the ED model were carriedout for each growth data set. As the exponential model is nestedin the ED model, we were able to compare the fits of the twomodels and to check whether the lag phase is justified for VCEand OD data sets. For each data set, the ED model, which takesinto account the lag phase systematically, fits better thanthe exponential model (P < 5%). Therefore, a significantlag phase exists for OD- and VCE-based data sets.

Fits of the Baranyi, Hills, and ED models.
The Baranyi, Hills, and ED models were fitted to both the VCEand OD growth data sets. All of these models proved to be effectivein modeling growth curves (Fig. 1 shows an example), and noneof them could be invalidated. RMSEs obtained from these fitsare presented in Table 2. An analysis of variance test for randomizedblock design was carried out with the RMSE values. This testindicated that in terms of RMSE, there were no significant differencesamong the fits of the three models for VCE data (P = 0.39) andfor OD data (P = 0.20). Consequently, although the ED modelfitted the best in 6 of the 12 cases for VCE data and in 5 ofthe 12 cases for OD data, none of the models consistently producedthe best fit to all the growth curves. Analysis of the Beale95% confidence regions (Fig. 2 shows an example) revealed animportant autocorrelation between ${lambda}$ and µ_max, especiallyfor the Baranyi and Hills models. The greatest precision inthe estimates of x₀, ${lambda}$ , and µ_max was obtained when theED model was fitted.

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FIG. 1. Fit for the strain IV 512 data set with the Baranyi, Hills, and ED models. Three corresponding scatter plots of normed residuals are presented on the right.

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TABLE 2. RMSE obtained after fitting of the Baranyi, Hills, and ED models

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FIG. 2. Plots of 95% confidence regions for estimated growth parameters obtained by fitting the Baranyi, Hills, and ED models (strain IV 512).

Estimates of ${lambda}$ were obtained by fitting the three growth models.Figure 3 shows box plots of the estimates from OD and VCE data.Furthermore, the results show that there were significant differencesin the estimates of ${lambda}$ provided by the three models for VCE data(P = 4 x 10^-11) and for OD data (P = 3 x 10^-7). As far as ourdata were concerned, we obtained systematically higher estimatesof ${lambda}$ when the Baranyi model was fitted, whereas the lowest estimateof this parameter was obtained when the ED model was fitted.We presumed that the bias might be all the more important sincethe ${lambda}$ was short. In addition, there were no significant differencesbetween ${lambda}$ estimates obtained from OD and VCE data for the Baranyimodel (P = 0.50), the Hills model (P = 0.50), or the ED model(P = 0.51).

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FIG. 3. Box plots of the estimates of ${lambda}$ obtained by fitting the Baranyi, Hills, and ED models to VCE (a) and OD (b) data sets. The 25th, 50th, and 75th percentiles and extreme values are shown.

Similarly, growth rate estimates are shown in Fig. 4. The resultsshow that there were significant differences in growth rateestimates among the three models for VCE data (P = 3 x 10^-8)and for OD data (P = 4 x 10^-8). While we found that comparedwith the other models the Baranyi model tended to systematicallygive higher estimates of ${lambda}$ , we demonstrated that it also gavehigher estimates of µ_max. As for the lag, there were nosignificant differences between the estimates obtained fromthe two kinds of data for the Baranyi model (P = 0.69), theHills model (P = 0.66), and the ED model (P = 0.35).

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FIG. 4. Box plots of the estimates of µ_max obtained by fitting the Baranyi, Hills, and ED models to VCE (a) and OD (b) data sets. The 25th, 50th, and 75th percentiles and extreme values are shown.

When we compared globally the estimates of ${lambda}$ based on OD andVCE data, we found no significant differences. However, individualcomparisons sometimes revealed discrepancies between the estimatesof ${lambda}$ based on OD and VCE data. Hudson and Mott (14) found that ${lambda}$ estimates based on OD data were systematically smaller than ${lambda}$ estimates based on VCE data. These authors explained theirresults by the existence of cell inflation during the lag phase.In our study, we showed that there was no systematic one-sidedbias in estimates of ${lambda}$ (Fig. 5).

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FIG. 5. Deviation in the estimates of ${lambda}$ based on OD and VCE data obtained by fitting the Baranyi, Hills, and ED models. The ${lambda}$ estimates obtained from each model are circumscribed.

Evolution of the OD/VCE ratio over time.
The next step in our study consisted of trying to understandwhy the variations in the ${lambda}$ estimates occur. We studied the evolutionof the logarithm of the OD/VCE ratio for the first phases ofgrowth (Fig. 6). We managed to get information concerning thevariation in the OD per cell unit.

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FIG. 6. Plot of log OD/VCE against time for the 12 strains.

Our results show that the log OD/VCE values are particularlyconstant around -8.9. However, in a few cases, we detected aslight linear increase in log OD/VCE from a lower value to roughly-8.9 (e.g., for strain IV 576, IV 634, or III 10126b). As theVCE value remained constant during the lag phase, this increasein the ratio was substantially due to an increase in the OD.Indeed, the cells for which we observed a visible increase hada lower absorbance around zero time. The increase in OD couldbe explained either by an increase in the cell biovolume orby an increase in the cell refraction.

Global modeling of the increase in OD/VCE during the lag phase.
We fitted a global model (equations 5 and 6) for each pair ofOD and VCE data sets. The full model accounts for an increasein the OD during the lag phase, whereas no increase in the ODis modeled in the partial model. As previously demonstrated,the ED model proved to have particularly good statistical properties.As a result, we chose to model x_VCE by using the ED model (ffunction in equations 5 and 6). The log-transformed global modelsused to fit our log-transformed data can be expressed as describedbelow.

The equations for the partial model are as follows:

(7)
where y₀ and ${alpha}$ are the decimal logarithms of x₀(initial VCE) and k (the x_OD/x_VCE ratio), respectively.

The equations for the full model are as follows:

(8)

We fitted the partial model and the full model to our experimentaldata and compared the fits of the two nested models by usingan F test. The results showed that for 6 of 12 data sets thefull model fitted significantly better than the partial model(Table 3).

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TABLE 3. Parameter estimates obtained by fitting the global model to the experimental data (simultaneously obtained OD and VCE data) and P values obtained with the nested model F test

By constructing and fitting our global model, we found thatthe evolution of OD exhibits different patterns (Fig. 7). Insome cases, there is no increase in OD during the lag phase.In other cases, one can detect an exponential increase in OD,whose intensity varies according to the strain and accordingto the experiment.

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FIG. 7. Fit of the global model for three pairs of data sets, the data sets for strains IV 644 (a), IV 646 (b) and III 10126b (c). For reasons of clarity, the two plots were superimposed by subtracting ${alpha}$ from the log OD data. Parameter estimates for the global model are also shown.

DISCUSSION

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Discussion

The aim of this study was to compare the estimates of ${lambda}$ obtainedby fitting different classical growth models (Baranyi, Hills,and ED models) to different kinds of data (OD and VCE data).In terms of RMSE, we could not invalidate any model. Indeed,none of the models fitted systematically better than the others.However, our results show that there are rather important differencesamong the estimates of ${lambda}$ when different models are used to fitdata. In particular, the Baranyi model gives high estimatesof ${lambda}$ (on average 46% higher for OD data and 49% higher for VCEdata than the Hills model and 80% higher for OD and VCE datathan the ED model). As a result, the choice of the model fittedto estimate ${lambda}$ appears to be crucial. The technique used to monitorbacterial growth seems to be less influential. Nevertheless,by analyzing strain growth curves one by one with OD and VCEdata, we found that sometimes there is a large difference between ${lambda}$ estimates based on the two kinds of data. By simultaneouslymonitoring the evolution of VCE and OD, we managed to find anincrease in OD due not to an increase in cell number but ratherto an increase in mean cell volume or to an increase in a meancell refraction.

We showed that the increase was not systematic and that thisphenomenon occurred in only 50% of our experiments. It is alsoworth noting that this phenomenon is not strain dependent. Indeed,the same strain used twice in experiments may not have the samegrowth pattern. Consequently, this growth behavior seems notto be due to interstrain variability but rather to an importantpopulation sensibility to growth and pregrowth conditions. Withregard to L. monocytogenes, we found that the mean log OD/VCEvalue measured at the end of the lag phase and during the exponentialphase is constant, whereas the first values measured at thebeginning of the lag phase tend to be lower for strains thatshow an increase in the OD during the lag phase. Thus, by determiningthe initial values of the OD/VCE ratio, one could predict whetheran increase in OD is likely. Additional data are necessary toconfirm these assumptions. However, we assume that predictingthis phenomenon is essential. When this phenomenon is not takeninto account, lower estimates of ${lambda}$ may be obtained with OD datathan with VCE data, as we observed, for example, with the Baranyimodel (Fig. 8).

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FIG. 8. ${lambda}$ estimates obtained by fitting the Baranyi model individually to OD and VCE data for the six strains for which significant increases in OD were detected.

The increase in OD reflects what happens at the cell level.If it is due to an increase in mean cell volume, the data maysuggest that cells need to reach an appropriate volume priorto initiating their first division (14). On the other hand,if the increase in OD is due to an increase in mean cell refraction,the data may suggest that cells need to accumulate intracellularsubstances necessary for growth (5) and tend to absorb lightmore intensively. In the latter case, dynamic study of the evolutionof the OD/VCE ratio can be related to a method which allowsus to monitor the evolution of the main internal molecules (essentiallyDNA, RNA, and proteins). As previously mentioned by authorsworking on the stochastic aspects of the lag phase (3, 16),new approaches based on analysis of individual cells (microscopy,flow cytometry, etc.) should help us explain the phenomenonobserved and, more generally, improve our understanding of thelag phase.

ACKNOWLEDGMENTS

We thank G. Fardel for technical assistance. We are also verygrateful to M. McInnes and N. Harris, who checked the manuscriptfor correct use of English.

FOOTNOTES

* Corresponding author. Mailing address: CNRS UMR 5558, Laboratoire de Bactériologie, Faculté de Médecine Lyon-Sud, BP 12, 69921 Oullins Cedex, France. Phone: 33-4-7886-3167. Fax: 33-4-7886-3149. E-mail: baty{at}biomserv.univ-lyon1.fr.

REFERENCES

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