A combined discreteâcontinuous model describing the lag phase of ...

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172 R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 function; however, there are some limitations to the Some of the fundamentals of this approach have use of this type of function. For example, it can be been discussed by McMeekin et al. (1993). These shown mathematically that the Gompertz rate is authors and others emphasized the limits of this always the maximum rate and occurs at an arbitrary method, the most severe of which is the fact that OD point of inflection (Garthright, 1991, 1997). The methods are comparative only, and cannot be used to Gompertz equation tends to overestimate m as it fits predict viable counts unless some attempt at calia sigmoidal curve to a straight line. In addition, the l bration is made (McMeekin et al., 1993; Baranyi and calculated with the Gompertz is always at a defined Roberts, 1995). McClure et al. (1993) used a simple point relative to the upper and lower asymptotes quadratic equation to relate OD to viable counts. (Garthright, 1991, 1997). Thus, empirical equations Dalgaard et al. (1994) used two equivalent methods have a limited ability to enhance our knowledge for calibration: one in which stationary phase cells concerning the physiological stages of bacterial were diluted to the appropriate OD, and the other in adaptation to new environment and subsequent which samples for OD and viable count were taken growth. during growth. Predicted generation times were It has been suggested that connecting the behavior lower with viable count data (Dalgaard et al., 1994), of a single cell to that of the whole population is the and this factor has been taken into account in later next stage in developing a more mechanistic ap- studies (Miles et al., 1997). Similar methods have proach to predictive food microbiology (Baranyi, been used to relate turbidimetric and viable count 1997). While there have been some attempts to data (Chorin et al., 1997). develop mechanistic models for bacterial growth In some studies, the Gompertz equation was fitted (Baranyi and Roberts, 1994, 1995; Hills and Wright, directly to OD data; however, no data was available 1994; Hills and Mackey, 1995), these models tend to 7 at below the minimum detectable OD (ca. 10 cfu/ view bacteria as a homogeneous population, and ml) thus the estimates for l and m should be there have been few attempts to model bacterial questioned (Hudson, 1994; Hudson and Mott, 1994). adaptation and growth on the basis of single cells. A form of calibration was achieved by relating l Recently, a model was proposed in which the using OD measurements to that determined with bacterial population was divided into non-growing viable counts by a regression equation (Hudson and and growing cells (McKellar, 1997). This model was Mott, 1994). McMeekin et al. (1993) have discussed expressed in the form of differential equations, and the correct way to fit the Gompertz function to % the behavior of the two types of cells was modeled transmittance data, and this method has been used to independently. Buchanan et al. (1997) have proposed calculate generation times (Neumeyer et al., 1997). a model which takes into account the variation in Other studies have been carried out without any adaptation (or lag) time of individual cells. Simula- apparent calibration (Huchet et al., 1995). l values tions with this model gave rise to ‘‘traditional’’ have been estimated from OD data by extrapolation growth curves; however, these authors did not pro- of the exponential portion of the curve back to the vide experimental evidence for their model. More initial cell numbers (Breand et al., 1997); however, recently, Baranyi and Pin (1999) and Baranyi (1998) this method may be inaccurate since the growth rate have proposed a lag model based on behavior of estimated from the OD data may be lower than that individual cells. obtained during the period of maximum growth Construction of models using viable count data is (McMeekin et al., 1993). time consuming and expensive, and researchers have Interestingly, the time to detection (t d) approach explored other, more rapid, methods for accumulat- has not been used to any great extent. The td for a ing sufficient data for modeling. One of the simplest turbidimetric instrument can be defined as the time method is the use of optical density (OD), where required for an initial measurable increase in OD. growth can be related to the increase in turbidity of a The difference between td for serial two-fold dilu- bacterial culture. This method lends itself particu- tions gives the doubling time, from which the m can larly well to automation, and a number of studies be determined (Cuppers and Smelt, 1993). The l can have used automated turbidmetric instruments such be calculated subsequently by the difference between as the Bioscreen (McClure et al., 1993; Huchet et al., the predicted td based on the m, and the observed td 1995). (Cuppers and Smelt, 1993). This method was used to
R.C. McKellar, K. Knight / International Journal of Food Microbiology 54 (2000) 171 –180 173 estimate individual cell lag times (Pin and Baranyi, t (h), that is, the time required for the Bioscreen to d 1998). Given the limitations and inherent inac- record a 0.05 increase in optical density from t . 0 curacies of the calibration method, the td approach Duplicate wells for each dilution were used for the would seem to be the only valid one. preparation of a standard curve of log OD against The purpose of the present study was to (1) obtain cfu/ml. For the determination of m, five replicate data on the lag phase experienced by single cells of wells were used for each dilution, with 20 replicate Listeria monocytogenes using the Bioscreen and (2) wells being used for dilutions which were close to 0 develop a discrete–continuous model which com- cfu/ml. bines cell adaptation as a property of individual cells Viable cells were enumerated for each two-fold (discrete activity) with a continuous model for dilution by spread plating 0.1 ml of appropriate serial bacterial growth. dilutions in duplicate onto tryptic soy agar (TSA; Difco Labs.). The plates were incubated at 308C for 48 h and colonies were counted using a Quebec Colony Counter (Model 15; American Optical, Buf- 2. Materials and methods 2.1. Strains and culture conditions Dynamic models were created using © ModelMaker Version 3.0 (Cherwell Scientific Pub- lishing, 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 Soft- ware, Salt Lake City, UT, USA). A modified Powell algorithm was used to minimize the sum of squared deviation between observed data and model calcula- tions. 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. 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 308C 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 2 258C. The contents of one cyrovial was transferred to 10 ml of TSB, incubated for 24 h at 308C 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. Identifi- cation Strip (BioMerieux Canada, St. Laurent, Canada) was used to confirm the identity of the culture. 2.2. Bioscreen growth experiments falo, NY, USA). 2.3. Modeling Serial two-fold dilutions of the inoculum were 3. Results made using fresh TSB to obtain a range of dilutions 5 representing approximately 10 to 0 cfu/ml. From Kinetic parameters describing bacterial growth can each of the two-fold dilutions, 0.35 ml was trans- be determined from turbidity data (Cuppers and ferred to wells of a Bioscreen plate (Labsystems, Smelt, 1993). Plots of td obtained from serial dilu- Helsinki, Finland). The filled plates were placed in tions of the original inoculum against ln cfu/ml (Fig. the Bioscreen (Labsystems) at an incubation temslope 1) gave straight lines, and m was calculated from the perature of 308C. Measurements were taken using a by Eq. (1): wide band filter, with pre-shaking at medium intensity for 10 s prior to OD reading; measurements were 1 m 52 ]] taken every 4 min for 25 h. Results were reported as Slope (1)
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