Scientific Publications - Work Done by Microbiology Reader Bioscreen C

International Journal of Food Microbiology, 2000, vol. 54, pp. 171-180

A combined  discrete–continuous  model describing the  lag phase of Listeria monocytogenes

R. C. McKellar and K. Knight

Southern Crop Protection and Food Research Centre – Food Research Program, Agriculture and Agri-Food Canada, 93 Stone Road West, Guelph, Ontario N1G 5C9, Canada

Received 30 April 1999; revised 9 August 1999; accepted 13 November 1999. Available online 8 March 2000.

 

ABSTRACT

Food microbiologists generally use continuous sigmoidal functions such as the empirical Gompertz equation to obtain the kinetic parameters specific growth rate () and lag phase duration () from bacterial growth curves. This approach yields reliable information on ; however, values for are difficult to determine accurately due, in part, to our poor understanding of the physiological events taking place during adaptation of cells to new environments. Existing models also assume a homogeneous population of cells, thus there is a need to develop discrete event models which can account for the behavior of individual cells. Time to detection (t_d) values were determined for Listeria monocytogenes using an automated turbidimetric instrument, and used to calculate . Mean individual cell lag times (t_L) were calculated as the difference between the observed t_d and the theoretical value estimated using . Variability in t_L for individual cells in replicate wells was estimated using serial dilutions. A discrete stochastic model was applied to the individual cells, and combined with a deterministic population-level growth model. This discrete–continuous model incorporating t_L and the variability in t_L (expressed as standard deviation; S.D._L) predicted a reduced variability between wells with increased number of cells per well, in agreement with experimental findings. By combining the discrete adaptation step with a continuous growth function it was possible to generate a model which accurately described the transition from lag to exponential phase. This new model may serve as a useful tool for describing individual cell behavior, and thus increasing our knowledge of events occurring during the lag phase.

Author Keywords: Lag; Listeria monocytogenes; Predictive modeling; Bioscreen; Turbidimetric; Discrete; Stochastic; Deterministic

1. INTRODUCTION

Bacterial growth data is normally analyzed using an empirical sigmoidal function such as the Gompertz equation (Gibson and Willocx). Useful kinetic parameters such as the maximum specific growth rate () and the lag phase duration () can be obtained from the Gompertz function; however, there are some limitations to the use of this type of function. For example, it can be shown mathematically that the Gompertz rate is always the maximum rate and occurs at an arbitrary point of inflection (Garthright and Garthright). The Gompertz equation tends to overestimate as it fits a sigmoidal curve to a straight line. In addition, the calculated with the Gompertz is always at a defined point relative to the upper and lower asymptotes (Garthright and Garthright). Thus, empirical equations have a limited ability to enhance our knowledge concerning the physiological stages of bacterial adaptation to new environment and subsequent growth.

It has been suggested that connecting the behavior of a single cell to that of the whole population is the next stage in developing a more mechanistic approach to predictive food microbiology(Baranyi, 1997). While there have been some attempts to develop mechanistic models for bacterial growth ( Baranyi; Baranyi; Hills and Hills), these models tend to view bacteria as a homogeneous population, and there have been few attempts to model bacterial adaptation and growth on the basis of single cells. Recently, a model was proposed in which the bacterial population was divided into non-growing and growing cells ( McKellar, 1997). This model was expressed in the form of differential equations, and the behavior of the two types of cells was modeled independently. Buchanan et al. (1997) have proposed a model which takes into account the variation in adaptation (or lag) time of individual cells. Simulations with this model gave rise to "traditional" growth curves; however, these authors did not provide experimental evidence for their model. More recently, Baranyi and Pin (1999) and Baranyi (1998) have proposed a lag model based on behavior of individual cells.

Construction of models using viable count data is time consuming and expensive, and researchers have explored other, more rapid, methods for accumulating sufficient data for modeling. One of the simplest method is the use of optical density (OD), where growth can be related to the increase in turbidity of a bacterial culture. This method lends itself particularly well to automation, and a number of studies have used automated turbidmetric instruments such as the Bioscreen (McClure and Huchet).

Some of the fundamentals of this approach have been discussedby McMeekin et al. (1993). These authors and others emphasized the limits of this method, the most severe of which is the fact that OD methods are comparative only, and cannot be used to predict viable counts unless some attempt at calibration is made ( McMeekin and Baranyi). McClure et al. (1993) used a simple quadratic equation to relate OD to viable counts. Dalgaard et al. (1994) used two equivalent methods for calibration: one in which stationary phase cells were diluted to the appropriate OD, and the other in which samples for OD and viable count were taken during growth. Predicted generation times were lower with viable count data ( Dalgaard et al., 1994), and this factor has been taken into account in later studies ( Miles et al., 1997). Similar methods have been used to relate turbidimetric and viable count data ( Chorin et al., 1997).

In some studies, the Gompertz equation was fitted directly to OD data; however, no data was available at below the minimum detectable OD (ca. 10⁷ cfu/ml) thus the estimates for and should be questioned (Hudson and Hudson). A form of calibration was achieved by relating using OD measurements to that determined with viable counts by a regression equation (Hudson and Mott, 1994). McMeekin et al. (1993) have discussed the correct way to fit the Gompertz function to % transmittance data, and this method has been used to calculate generation times ( Neumeyer et al., 1997).

Other studies have been carried out without any apparent calibration (Huchet et al., 1995). values have been estimated from OD data by extrapolation of the exponential portion of the curve back to the initial cell numbers (Breand et al., 1997); however, this method may be inaccurate since the growth rate estimated from the OD data may be lower than that obtained during the period of maximum growth ( McMeekin et al., 1993).

Interestingly, the time to detection (t_d) approach has not been used to any great extent. The t_d for a turbidimetric instrument can be defined as the time required for an initial measurable increase in OD. The difference between t_d for serial two-fold dilutions gives the doubling time, from which the can be determined (Cuppers and Smelt, 1993). The can be calculated subsequently by the difference between the predicted t_d based on the , and the observed t_d (Cuppers and Smelt, 1993). This method was used to estimate individual cell lag times ( Pin and Baranyi, 1998). Given the limitations and inherent inaccuracies of the calibration method, the t_d approach would seem to be the only valid one.

The purpose of the present study was to (1) obtain data on the lag phase experienced by single cells of Listeria monocytogenes using the Bioscreen and (2) develop a discrete–continuous model which combines cell adaptation as a property of individual cells (discrete activity) with a continuous model for bacterial growth.

2. MATERIALS AND METHODS

2.1. Strains and culture conditions

Listeria monocytogenes Scott A (human clinical isolate) was obtained from the culture collection at the Food Research Program (Guelph, Canada). The culture was grown for 24 h at 30°C in tryptic soy broth (TSB; Difco Labs., Detroit, MI, USA). Stock cultures were prepared in TSB plus 15% glycerol (BDH, Toronto, Canada) and were frozen in 0.3-ml aliquots in cyrovials at −25°C.

The contents of one cyrovial was transferred to 10 ml of TSB, incubated for 24 h at 30°C in a shaking waterbath (New Brunswick Scientific, Edison, NJ, USA) at 1500 rpm. The culture was transferred (1%) to 10 ml fresh TSB and incubated under the same conditions. The resulting culture was used as the inoculum for experiments. API Listeria spp. Identification Strip (BioMerieux Canada, St. Laurent, Canada) was used to confirm the identity of the culture.

2.2. Bioscreen growth experiments

Serial two-fold dilutions of the inoculum were made using fresh TSB to obtain a range of dilutions representing approximately 10⁵ to 0 cfu/ml. From each of the two-fold dilutions, 0.35 ml was transferred to wells of a Bioscreen plate (Labsystems, Helsinki, Finland). The filled plates were placed in the Bioscreen (Labsystems) at an incubation temperature of 30°C. Measurements were taken using a wide band filter, with pre-shaking at medium intensity for 10 s prior to OD reading; measurements were taken every 4 min for 25 h. Results were reported as t_d (h), that is, the time required for the Bioscreen to record a 0.05 increase in optical density from t₀.

Duplicate wells for each dilution were used for the preparation of a standard curve of log OD against cfu/ml. For the determination of , five replicate wells were used for each dilution, with 20 replicate wells being used for dilutions which were close to 0 cfu/ml.

Viable cells were enumerated for each two-fold dilution by spread plating 0.1 ml of appropriate serial dilutions in duplicate onto tryptic soy agar (TSA; Difco Labs.). The plates were incubated at 30°C for 48 h and colonies were counted using a Quebec Colony Counter (Model 15; American Optical, Buffalo, NY, USA).

2.3. Modeling

Dynamic models were created using ModelMaker^© Version 3.0 (Cherwell Scientific Publishing, Oxford, UK).

The Gompertz and heterogeneous population (HPM) (McKellar, 1997) models were fit to growth data (obtained either from actual or simulated cell counts) using Scientist^® (Micromath Scientific Software, Salt Lake City, UT, USA). A modified Powell algorithm was used to minimize the sum of squared deviation between observed data and model calculations. Initial parameter estimates were obtained using simplex optimization. Differential equations were solved numerically by the method of Runge–Kutta, since the software does not require analytical forms of equations.

Prism Version 2.0 (GraphPad Software for Intuitive Science, San Diego, CA, USA) was used to create plots.

3. RESULTS

Kinetic parameters describing bacterial growth can be determined from turbidity data (Cuppers and Smelt, 1993). Plots of t_d obtained from serial dilutions of the original inoculum against ln cfu/ml (Fig. 1) gave straight lines, and was calculated from the slope by Eq. (1):

Fig. 1. Determination of specific growth rate () and lag phase duration (t_L) for Listeria monocytogenes using time to detection (t_d) data obtained from the Bioscreen. Experimental data (•), simulated data ().

 

It is also possible to calculate from cell counts obtained by using either quadratic (McClure et al., 1993) or cubic ( Stephens et al., 1997) calibration curves to convert Bioscreen absorbance data; however, this method was not employed in the present study.

Calculation of using a serial dilution method is independent of the absolute numbers of cells present. However, it is more difficult to calculate the individual cell lag phase duration (t_L). It was assumed that the dilution giving the largest t_d was equal to ln cfu/well=0 (Fig. 1). The calculated was used in the HPM to predict the time required to detect growth from a defined number of cells where the detection limit of the Bioscreen is 3.5·10⁶ cfu/well. The detection limit was confirmed by means of a calibration curve (data not shown). Fig. 1 shows that simulated values for t_d underestimated the experimental t_d by an amount equivalent to t_L. Note that for each dilution, t_L was constant, thus was independent of cell numbers. Replicate values of t_L were calculated from 20 wells by subtracting the simulated value for t_d from the replicate experimental values, and the resulting mean t_L and standard deviations (S.D._L) are given in Table 1 for two trials. S.D._L values are based on <20 wells giving growth; in the two trials reported here 12 and 13 wells, respectively, showed growth.

 

 

Table 1. Kinetic parameters for Listeria monocytogenes determined using the Bioscreen^a

 

The S.D._L values in trial A (Table 1) are based on the supposition that all 12 wells showing growth arise from a single cell. The probability of finding a single cell per well can be calculated from a Poisson distribution:

where P(X=i) is the probability of finding i cells in a randomly chosen well, and is the expected value of that cell number.

Using the observation that 12/20 or 60% of wells contain one or more cells, the value of may be calculated from the following equation:

Substituting (0.916) in Eq. (2), it is possible to calculate the probability of finding one (37%) or two (17%) cells per well. Thus, as many as five of the 12 wells showing growth could have arisen from more than one cell. This suggests that the S.D._L values must be considered only as estimates for single cells. A more direct method (such as microscopic examination) is needed to obtain accurate distributions of single cell t_L values.

The simulation software, ModelMaker^©, was used to develop a combined discrete–continuous model which can account for the behavior of individual cells, and is described in the diagram in Fig. 2. Note that the various blocks in Fig. 2 are of different shape depending on their function: compartment blocks are rectangular, and the values change with time according to user-defined differential equations; variable blocks have rounded ends, and values are calculated at each time interval according to user-defined explicit equations; defined value blocks have pointed ends, and values are assigned at t₀ or at particular times during the simulation; independent event blocks are hexagonal, and are activated at a pre-defined time; and component event blocks are square, and are activated in response to other components in the model.

 

 

Fig. 2. Discrete–continuous model designed with ModelMaker^© to simulate t_d and standard deviation of individual cell lag times (S.D._L) based on kinetic parameters derived from Bioscreen data.

 

When the model run is initiated, the AssignLag block (an independent event block activated at t₀) uses a random number generator based on a truncated (positive values only) normal distribution with mean t_L and S.D._L from Table 1 to assign t_L values to each of up to 64 cells. These values are stored in the defined value Triggers block. The Adaptation block reads these values, and adds a single cell to the Growing compartment at each time corresponding to an individual cell t_L. Once in the Growing compartment, cells start growing immediately according to a logistic equation (McKellar, 1997)

where G is the number of cells in the Growing compartment and N_max is the maximum population density.

The LogGrowing block calculates the log cfu, and at each time increment the component event Monitor block tests to see if the value of LogGrowing has exceeded the detection limit (defined as 3.5·10⁶ cfu/well). The defined value TimetoDetection block holds the calculated t_d.

The model in Fig. 2 was used to simulate values for t_d corresponding to up to 32 cells per well. The simulated S.D. values were derived from a total of 20 simulations for each of 1, 2, 4, 8, 16 or 32 cells per well. Fig. 3 shows that both the simulated t_d and S.D. values are in close agreement with the experimental findings. Note that S.D._L refers to the variation in individual cell lag times, t_L, whereas S.D. refers to the variation between wells observed with any defined number of cells per well.

 

 

Fig. 3. Comparison of t_d () and S.D._L () from experimental data (open symbols) or simulated data (closed symbols).

 

The model described in Fig. 2 was extended to allow the simulation of a complete growth curve. The discrete adaptation function was retained, and combined with the HPM to provide the continuous growth function ( Fig. 4). In this model, the Adaptation block moves one cell from the NonGrowing to the Growing compartment at each of the Trigger times. This preserves the initial number of cells in the model. The blocks to calculate the log of the cell numbers are provided to assist in the visualization of the growth curve.

 

 

Fig. 4. Discrete–continuous model designed with ModelMaker^© to simulate a complete bacterial growth curve.

 

The output of this model for a total of 64 cells is shown in Fig. 5. Note that values of the the Growing compartment are not shown where the number of cells was zero. These results show that the simulated values for the NonGrowing cells decreased as the numbers of Growing cells increased. When the total number of cells in the model was calculated, the resulting curve represents the transition from the lag phase to the exponential phase.

 

 

Fig. 5. Output of discrete–continuous model showing number of cells present in the NonGrowing () and Growing (•) compartments, and the total of both compartments ().

 

The ability of the discrete–continuous model to simulate actual growth curves was further tested by comparing the simulated output with growth curves derived from plate count data. Simulated growth curves were created using the values for t_L and S.D._L from Table 1 (with N₀ and N_max being set at 64 and 10⁹ cells, respectively), and were fit with the Gompertz and the HPM (Table 2, trials A and B). Experimental growth curves at 30°C were also fit with the Gompertz and the HPM (Table 2, trials C and D). The reference trial is from a previous study ( McKellar, 1997). The results in Table 2 show good agreement between the simulated growth curves based on Bioscreen data, and the actual growth curves derived from plate count data. Values for were lower with the HPM (0.918–1.09) as compared to the Gompertz (1.03–1.26), and were similar for Bioscreen (0.918–1.26) and viable count data (1.03–1.26). The values were larger with the Bioscreen as compared to viable counts, with the HPM giving shorter times than the Gompertz with the exception of Trial D. Fig. 6 shows a comparison between one experimental and one simulated growth curve both fit with the Gompertz function. When discussing lag times, it should be noted that the symbol is reserved for the population lag phase duration calculated from growth curves (either viable count or simulated) using a non-linear curve fitting routine, while the symbol t_L refers to the mean individual cell lag times determined using the Bioscreen.

 

 

Table 2. Kinetic parameters for Listeria monocytogenes determined using non-linear regression

 

Fig. 6. Comparison of simulated () and experimental (•) growth curves fit with the Gompertz function.

 

The influence of the S.D._L at constant t_L on the is shown in Fig. 7. When the S.D._L is small (open circles) the adaptation is rapid, as all the cells adapt at approximately the same time. As the S.D._L increases to a maximum (open triangles), the number of cells adapting earlier than the majority increases. These cells quickly dominate, and the result is a shorter .

 

 

Fig. 7. Effect on S.D._L on apparent population lag phase duration (), where S.D._L is: 0.0 (); 0.3 (•); 0.6 (); 0.9 (); and 1.2 ().

 

The influence of distribution type was also examined. Replicate simulations were performed with the discrete–continuous model using either a normal or an exponential (Baranyi, 1998) distribution for 20 replicate wells each containing 1, 2, 4, 8, 16 or 32 cells. A different random number seed was used for each replicate simulation. t_L and S.D._L values (normal distributions) were taken from trial A in Table 1, and t_L values for the exponential distributions were adjusted lower in order to separate the two data sets. Fig. 8 shows that, with an exponential distribution, the t_d values deviate from linearity when the number of cells per well <4. Mean individual cell t_d values (1 cell per well) varied considerably depending on the random number seed. In contrast, simulations using normal distributions were linear over the whole range of cell numbers, and random number seed had little apparent effect.

 

Fig. 8. Replicate simulations using the discrete–continuous model (Fig. 2) with three different random seed numbers for normal (open symbols) and exponential (closed symbols) distributions. t_d values were simulated for a total of 20 wells at each of 1, 2, 4, 8, 16 or 32 cells per well, with t_L and S.D._L values taken from trial A (Table 1).

 

4. DISCUSSION

A discrete–continuous model has been developed which improves our understanding of the behavior of cells during the period of adaptation generally referred to as the "lag phase". In this model the lag phase is described by two parameters, the mean individual cell lag time, t_L, and the standard deviation of the variation between t_L values, S.D._L. When these two parameters are used with , N₀, and N_max, a complete growth curve can be simulated. It should be noted that a key assumption of this model is that the t_L is independent of cell number. The substance of the present study was presented previously (McKellar, 1998).

The present study also reports, for the first time, the inclusion of a discrete step into the modeling of bacterial growth; all other published models are based on continuous functions only. This improvement is critical to the modeling of single cell behavior during the adaptation period prior to initiation of growth, and is facilitated by the use of an object-oriented programing environment such as ModelMaker^®. Individual-based models (IBMs) have been used extensively in ecological modeling situations (Grimm and Lomnicki), but have yet to be applied in food microbiology. Providing a dynamic environment for the simulation of bacterial growth based on the use of differential rather than explicit equations is also considered important for the future development of bacterial growth models ( Baranyi, 1997). Common software packages which are generally used for non-linear regression do not have this capability, thus the use of software such as ModelMaker^® leads to the development of more complex models incorporating the multiple steps involved in adaptation and growth.

A theoretical model which accounts for the behavior of individual cells has been suggested by Buchanan et al. (1997), who were the first to propose that the transition between lag and exponential phases resulted from biological variation among individual cells. These workers provided a theoretical basis for describing the lag phase in terms of individual cells; however, they proposed a simpler, three-phase model for general use which did not account for inter-cell variation. The present model builds on this foundation by the addition of turbidimetric data which provides evidence for individual cell behavior, and incorporates this variability into the model as a distinct parameter (S.D._L). Buchanan et al. (1997) used the risk analysis software, @RISK™, to simulate the variability between cells, but fit only the three-phase linear model to experimental data. Since the present model contains a random number generator, it was not possible to fit experimental data using the optimization algorithms in ModelMaker^®, thus optimization must be performed manually. It will be possible, however, to construct tables of distributions with varied S.D._L which can be read into the model. This will allow limited fitting of the discrete–continuous model to experimental data, and will form the basis of a later publication.

Baranyi and Pin (1999) have also recently proposed a method for calculating and from t_d. Their method is based on the biological interpretation of the initial physiological state of the cells, where the suitability for growth is represented by a fraction of the initial cell population. This interpretation is similar to the one suggested by McKellar (1997) who attributed the potential for growth to a sub-population of the inoculum. The Baranyi and Pin approach uses an analysis of variance (ANOVA) method to deal with variability of low cell populations to estimate a value for . Values for t_L are calculated using the and the physiological state of the inoculum. In the present study, values for are estimated using a wider range of dilutions than reported by Baranyi and Pin, thus minimizing the influence of higher variance.

Buchanan et al. (1997) describe the lag phase by the following equation:

where t_a is the time required for the cells to adapt to their new environment, and t_m is the generation time. The present model assumes that growth starts immediately after the adaptation step, thus t_L (this study) is equivalent to t_a.

Baranyi (1998) has recently compared stochastic and deterministic concepts of the lag phase, and has suggested that the is always less than the t_L. This seems reasonable, since increased S.D._L at constant t_L resulted in a shorter in the present study (Fig. 7) and also in the @RISK simulations reported by Buchanan et al. (1997). It is intuitively obvious that can only be equal to t_L in the special case where the cells all adapt simultaneously (e.g., S.D._L=0). In the present study using simulated growth curves, was greater than t_L when determined by the Gompertz function, and identical to t_L in one of two trials using the HPM. This may be due to the inherent error associated with estimations of from fitting data with non-linear regression functions. It is also worth mentioning that neither the Gompertz nor the HPM are intended for fitting data derived from distributions of individual cell properties, since neither of these models can account for changes in curvature between lag and exponential phases under the control of the S.D._L parameter. An example of this is evident in Fig. 6; the Gompertz function fit to a discrete–continuous model simulation shows systematic deviations. Systematic differences between predicted by several models including the three-phase linear model of Buchanan and the Baranyi model have been reported (Buchanan et al., 1997). These observations may even suggest that once individual cell behavior can be modeled accurately, the traditional concept of population lag () as a modeling parameter will be of limited further value to predictive microbiology.

It is difficult to compare the discrete–continuous model with the Baranyi model since the latter does not include a parameter describing the influence of variation in t_L on . Baranyi (1998) has suggested that the mean population lag [(N)] increases with lower inoculum, assuming t_L values are exponentially distributed. In the present study a normal distribution was assumed for t_L, and with that there was no evidence to support an increased t_L at lower inocula using either simulations or experimental results. Additional simulations using exponential rather than normal distributions for t_L resulted in predictions in which t_d values deviated from linearity, and changed depending on the random number seed. These simulations support the suggestion that t_L values are normally distributed. Further experiments are required to establish the correct distribution.

It must be emphasized that while the t_d decreases with increasing cell number, the t_L remains constant (Fig. 1). Similarly, S.D._L is constant, and the apparent decrease in variability between wells with increasing cell numbers per well is a consequence of the majority of the cells having a t_L close to the mean. As numbers of cells per well increase, the observed t_d for each well becomes less dependent on the t_d for individual cells, and starts to reflect the mean t_d value.

There are some limitations to the use of the Bioscreen for indirectly estimating single cell behavior. It is difficult to clearly identify the dilution corresponding to single cells; it is assumed to be the dilution giving the greatest average t_d. In addition, assuming that cells follow a Poisson distribution, some wells which are assumed to have single cells might have two or more cells. Wells which show no growth are assumed to have no cells; however, the possibility that some cells do not grow has not been considered. Thus the S.D._L must be considered an estimate of the value for single cell variability. In spite of this limitation, the discrete–continuous model based on estimated t_L and S.D._L values gives a good fit to the experimental data. More accurate estimates of single cell variance might be obtained using improved methods of single cell analysis (e.g., microscopy).

ACKNOWLEDGEMENTS

The authors would like to thank J. Baranyi for helpful criticism of the manuscript.

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