Scientific Publications - Work Done by Microbiology Reader Bioscreen C

International Journal of Food Microbiology, 2001, vol. 73, March 11 No.(2-3), pp. 137-144

Growth pH does not affect  the initial physiological state parameter ( p₀ ) of Listeria monocytogenes 

R. C. McKellar, X. Lu and K. P. Knight

Food Research Program, Agriculture and Agri-Food Canada, 93 Stone Road West, Guelph, Ontario, Canada N1G 5C9

Received 16 May 2001;  accepted 9 August 2001.  Available online 1 March 2002.

 

ABSTRACT

It has proven difficult to develop adequate mathematical models for the lag phase () which characterizes the adaptation period prior to the initiation of exponential growth by microorganisms. This is due, in part, to our incomplete understanding of the nature of the initial physiological state of cells (defined as h₀ or p₀ depending on the model), and changes taking place during adaptation. The objectives of the present study were to characterize p₀ using data from growth of Listeria monocytogenes in an automated turbidimetric instrument (Bioscreen), and to determine the influence of limiting growth pH. A model was developed for individual cells which combined a continuous adaptation phase (defined by p₀) with a discrete step marking the transition to a continuous exponential growth phase (the CDC model). Parameters of the new model were: p₀; the specific growth rate (); the initial cell number (N₀); and the maximum cell density (N_max). Progressive reduction of the growth pH in the Bioscreen to 4.7 decreased the . It was noted that the regression lines for all trials at all pH values appeared to have a common x-intercept (20.086±1.092), and it was deduced that, when the Bioscreen detection limit (15.07 ln cfu well⁻¹) was subtracted, the resulting value represented the "true" value for the initial physiological state of the cells.

Author Keywords: Mathematical model; Predictive microbiology; Lag phase; Growth; Listeria monocytogenes; Physiological state; pH
 

1. INTRODUCTION

Predictive microbiology, the use of mathematical models to describe the behaviour of foodborne microorganisms, has developed rapidly over the past decade, and has been of considerable value to the food industry (McMeekin et al., 1993). Most effective models to date have focussed on modelling the specific growth rate () of microorganisms, but attempts to model population lag phase duration () have been less successful (Augustin; Ross and Delignettemuller). The and have been shown to be related by a physiological state parameter (h₀; Eq. (1))

which represents the potential of the inoculum for growth, thus implying a constant relationship between the two parameters (Baranyi and Roberts, 1994). The h₀ has been shown to vary, however, due in part to the influence of preincubation conditions on the (Buchanan; Hudson; McClure; Gay and Dufrenne). Even when preincubation conditions were relatively constant, extensive variations in h₀ were observed when cells were subsequently cultured under conditions which limited growth (Delignettemuller; Robinson and Augustin). Thus, the significance of h₀ and its relationship to the physiological state of the cell population is still unclear.

The h₀ for a growing culture can be determined by fitting viable count data to the Baranyi model (Baranyi and Roberts, 1994) or the heterogeneous population model (McKellar, 1997). Unfortunately, the fitting of "traditional" growth curves will only give values of h₀ as a function of and . Thus, there is a clear need for an independent method for estimating the value of h₀ (Baranyi and Alavi).

Automated turbidimetric instruments such as the Bioscreen are gaining in popularity as a tool for developing kinetic growth models (Cuppers; Stephens; Baranyi and McKellar). Estimates of can be made with the Bioscreen (Baranyi and McKellar); however, there is presently no simple way of determining and h₀. The mean individual cell lag (t_L) can be calculated from Bioscreen data as the difference between the observed time to detection (t_d) for single cells, and the simulated t_d obtained using the calculated (McKellar and Knight, 2000). The mean individual cell physiological state (p₀) can then be calculated with Eq. (2).

Baranyi (1998) has proposed that the suitability of a population of cells for growth is the mean of the suitability of the individual cells; however, it is not clear if the p₀ calculated from Bioscreen data is equivalent to the population physiological state (h₀). Thus, the objectives of the present study are to (1) describe a simple, geometric method of calculating p₀ and t_L from Bioscreen t_d data, and (2) determine the influence of growth pH on p₀. A preliminary account of this work was presented previously (McKellar et al., 2000a).

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, Ontario, Canada). The culture was grown for 24 h at 30 °C in tryptic soy broth (TSB; Difco Laboratories, Detroit, MI). Stock cultures were prepared in TSB plus 15% glycerol (BDH, Toronto, ON) 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) at 150 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, PQ) 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 adjusted to the desired pH using HCl to obtain a range of dilutions representing approximately 10⁸–0 cfu ml⁻¹. 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 preshaking at medium intensity for 10 s prior to OD reading. Measurements were taken every 4 min for 25 h (pH 7.2 and 6.5), or every 20 min for 4 (pH 6.0, 5.5, and 5.0) or 5 (pH 4.9, 4.8, and 4.7) days. 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 well⁻¹. For the determination of , five replicate wells were used for each dilution, with 20 replicate wells being used for dilutions that were close to 0 cfu well⁻¹.

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). The plates were incubated at 30 °C for 48 h and colonies were enumerated using a Quebec Counter (American Optical, Model 15, Buffalo, NY).

2.3. Data analysis and modelling

Models were developed using ModelMaker© Version 3.0.3 (Cherwell Scientific Publishing, Oxford, UK). Linear regression was carried out using Prism™ 3.0 (GraphPad Software for Intuitive Science, San Diego, CA, USA).

3. RESULTS

The CDC model was designed in ModelMaker®, which has the advantage that it allows a combination of differential and explicit equations as well as discrete actions, required for the implementation of individual cell adaptation (Fig. 1). Note that the various blocks in Fig. 1 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; submodels are rectangular with double edges, and may contain groups of other blocks; independent event blocks are hexagonal, and are activated at a predefined time; and component event blocks are square, and are activated in response to other components in the model.

 

 
(14K)

Fig. 1. Structure of the midlevel module of the continuous–discrete–continuous (CDC) model describing the lag phase and exponential growth of L. monocytogenes. The model was constructed using ModelMaker Version 3.0.3. Description of the various blocks and their function is in the text.

 

Fig. 1 shows the midlevel module of the CDC model. The AssignState block (an independent event block activated at t₀) assigns up to 64 values for the initial individual cell physiological states using a random number generator which draws values from a truncated normal distribution with mean equal to a physiological state parameter (p₀) and variability expressed as standard deviation (SD_p0). These values are stored in an array of 64 compartments (CellState block in Fig. 1). An adaptation rate (set equal to [mu in Fig. 1]) is also assigned for each cell, and stored in an array of 64 defined values (AdaptRate block in Fig. 1). As the simulation progresses, the individual physiological states (p_i) values in the CellState array decrease with time at rate equal to .

The discrete adaptation step is implemented by four groups of 16 component event blocks (contained in the submodels CellGroupA, CellGroupB, CellGroupC, and CellGroupD), which monitor the values in the 64 CellState compartments. When the value of a CellState compartment <0, the corresponding Component Event block moves one cell from the NonGrowing compartment (where all cells were initially assigned at t₀) to the Growing compartment (Fig. 1). Cells in the Growing compartment start growing immediately with rate in response to a logistic function shown inEq. (3) (McKellar and McKellar):

where G is the number of cells in the growing compartment and N_max is the maximum cell concentration.

The log₁₀ of the total number of cells in the Growing compartment (G) and in the combined Growing and NonGrowing compartments (N) is calculated in variable blocks LogGrowing and LogTotalCells, respectively, to assist with visualization of the simulation results. Model parameters are: p₀, SD_p0 (standard deviation of p₀ taken from individual bioscreen wells; McKellar et al., 2000b), N₀ (initial total number of cells in the NonGrowing compartment); and N_max. The crosshatched blocks have values that are universal and can be used by any submodel.

The results of a sample simulation with =0.833 (corresponding to 30 °C), p₀=1.5, and SD_p0=0.6 are shown in Fig. 2. The dotted lines represent six of a total of 64 p_i values actually simulated in the model, the dashed line represents the value of the LogGrowing variable (from Fig. 1), and the solid line represents the value of the LogTotalCells variable (from Fig. 1). It should be noted that p₀ was selected as a positive value; thus, p_i values will decrease with time.

 

 

Fig. 2. Output from a simulation using the CDC model where the dotted lines represent six of a total of 64 individual physiological states (p_i; contents of the CellState block array in Fig. 1), the dashed line represents the value of the LogGrowing variable (Fig. 1), and the solid line represents the value of the LogTotalCells variable (Fig. 1). Parameter values used in the simulation are: specific growth rate ()=0.833 (corresponding to 30 °C), mean initial physiological state of the inoculum (p₀)=1.5, and the standard deviation of the mean initial physiological state (SD_p0)=0.6.

 

As a working hypothesis, p₀ was defined in the CDC model as the extent to which stationary phase genes are expressed; thus, adaptation can be thought of as a time-dependent reduction in the concentration of stationary phase gene products during adaptation to a new growth environment. In the Baranyi model, the physiological state parameter (h₀) is defined as a transformation of the initial concentration of a substance critical for growth which increases with time (Baranyi and Roberts, 1994). Since these two definitions appear quite different, it is necessary to compare them.

The relationship between p₀ and h₀ is shown in Fig. 3. The value for the initial p_i (or p₀ when a single cell is being considered) decreases on an ln scale to 0 at a time corresponding to the individual cell lag time (t_L) giving line "a", at which point the cell ‘reflects back’ in response to the discrete adaptation event to form an exponential growth curve (line "b"). If line "a" is rotated about the x-axis, line "c" is obtained (dotted line). This is now reminiscent of the HPM (McKellar, 1997), in which the intercept of the line describing the time-dependent value of the Growing compartment (here represented as lines "c" and "b" combined) with the y-axis denoted the ln of the initial number of growing cells (G₀). Note that when N₀=1, G₀ must represent <1 cell when t_L>0 (Fig. 3). G₀ is related to h₀ by Eq. (4) (McKellar, 1997).

Thus, for the special case of a single cell (where ln N₀=0), h₀=−ln G₀. It is obvious from Fig. 3 and the above argument that h₀ and p₀ are identical for a single cell.

 
 

Fig. 3. Output of a simulation for the adaptation and growth of a single cell using the CDC model. h₀, Baranyi physiological state parameter; t_L, individual cell lag time. Line "a" represents the change in the p_i from the initial physiological state (p₀), line "b" represents the exponential growth of a single cell, and line "c" represents the hypothetical "internal growth" of a single cell prior to the discrete adaptation step at t_L. The relationship between h₀ and p₀ is described in the text.

 

The predictive value of the CDC model depends to a large extent on a more complete understanding of p₀. In order to gain additional information on factors influencing the cell physiological state, we investigated the influence of decreasing growth pH on the t_d. The relationship between t_d and the initial number of cells per Bioscreen well is shown in Fig. 4 for pH values from 7.2 to 4.7. As the pH decreased, the slope of the regression lines increased (decreased ). At pH values <5.0, considerable variation in slope and position of the regression line was noted between trials. It was also observed that the maximum t_d at higher pH values (5.0–7.2) occurred at an ln cfu well⁻¹ of approximately 0, whereas at pH lower than 5.0 considerably greater numbers of cells were present in each well at the maximum t_d.

Fig. 4. Influence of growth medium pH on t_d for L. monocytogenes at various initial cell concentrations. Experiments at the various pH values are: 7.2 (), four reps; 6.5 (•), three reps; 6.0 (), four reps; 5.5 (), two reps; 5.0 (), two reps; 4.9 (), four reps; 4.8 (), three reps; 4.7 (), three reps.

 

It was observed from Fig. 4 that all regression lines appeared to intersect at a single point on the x-axis. This value was calculated to be 20.086±1.092 (SD) for a total of 24 Bioscreen experiments (replicate experiments were: four for pH 7.2, 6.0, and 4.9; three for pH 6.5, 4.8 and 4.7; and two for pH 5.5 and 5.0). It was tempting to speculate that this value was related in some way to the initial physiological state of the cells, and further analysis of the data revealed that this was indeed the case. We had previously shown that the mean individual cell lag (t_L) can be calculated from Bioscreen data as the difference between the observed t_d for single cells, and the simulated t_d obtained using the calculated (McKellar and Knight, 2000). This relationship is shown in Fig. 5. When experimental t_d data from the Bioscreen for a given pH (in this case, 7.2) were plotted against ln cfu well⁻¹ of L. monocytogenes determined by viable count, the solid line in Fig. 5 was obtained. From previous work (McKellar and Knight, 2000), it was apparent that simulated t_d values for a range of initial cell concentrations calculated using the slope of the experimental data and plotted against ln cfu well⁻¹ formed a parallel line (Fig. 5: broken line). The x-intercept of this line represents the detection limit of the Bioscreen (the concentration of cells needed to give a t_d=0), corresponding to 3.5×10⁶ cfu well⁻¹ (or ln cfu well⁻¹=15.07). This can be measured independently using a calibration curve. The t_L is then the difference between the y-intercepts of the experimental and simulated lines (Fig. 5). It has been shown that the population lag () and are related by Eq. (1) (Baranyi and Roberts, 1994) or Eq. (2) for the case of a single cell as described above. It is clear from Fig. 5 that the difference between the x-intercept of the experimental data and the detection limit of the Bioscreen (15.07) is equal to p₀, the initial physiological state of the inoculum. This can then be calculated as 5.016±1.092, based on the x-intercepts in Fig. 4.

 

 
(4K)

Fig. 5. Relationship between maximum specific growth rate (), mean individual lag phase duration (t_L), and initial physiological state (p₀) using time to detection (t_d) data from the Bioscreen.

 

Replicate values of p₀ were plotted against pH, with variability between replicate experiments at each pH expressed as SD and indicated by error bars, and the results with regression line are shown in Fig. 6. Regression analysis revealed that the slope of the regression line was not significantly different (p>0.05) from zero, thus p₀ was independent of the growth pH.

 

 
(3K)

Fig. 6. Relationship between initial physiological state (p₀) and pH. Bars show SD.

4. DISCUSSION

A series of discrete models have been developed to more clearly define the influence of p₀ on the growth of individual L. monocytogenes cells. An early version of a model to describe the adaptation and growth of individual cells was based on the mean individual cell lag times (t_L) and the variability of t_L expressed as standard deviation (SD_L) obtained from Bioscreen turbidimetric data (McKellar and Knight, 2000). This model was limited in that the individual lag times were preset at t₀, thus the model was not dynamic in the lag. The revised model incorporated a continuous adaptation step prior to the discrete shift from adaptation to exponential growth, and was thus able to handle non-isothermal conditions. This provided an improvement over existing dynamic models, which describe adaptation of homogeneous populations of cells (Alavi et al., 1999). In the present study, we have extended our understanding of the discrete model, and have shown the relationship between h₀ and p₀ (equal for single cells).

The improved model was still limited in that no method presently exists to independently estimate the value of p₀. At present, the model uses p₀ calculated from t_L using Bioscreen data, with the implicit assumption that p₀ is influenced only by t_L and . The present study provides further evidence for the validity of using Bioscreen data to directly measure p₀. The population physiological state (h₀) is determined directly from viable counts, and tracks only the changes in population lag phase duration () (McKellar and Baranyi). Bioscreen growth data (as in Fig. 4 and Fig. 5) represent "mirror images" of traditional growth curves; varied initial inoculum levels are allowed to grow for a fixed time (t_d) in the Bioscreen whereas traditional growth curves track cell numbers at different times from a single inoculum. Thus, the geometric definition of p₀ is equivalent to h₀ only in the special case of single cells, and p₀ describes the mean physiological state of the cells distinct from factors (as yet unknown) which might independently influence .

The advantage of the present approach can be seen in the influence of pH on growth of L. monocytogenes in the Bioscreen. Numerous publications have noted that the product of and (and thus h₀) is a constant (Griffiths; Mackey; McKellar; Delignettemuller; Giannuzzi; Robinson; Grijspeerdt; Martens and Molina). Some of these and other studies have shown that while a proportional relationship may exist between and , the values obtained are highly variable (Dufrenne; Delignettemuller; Robinson; Alavi and Augustin). A more detailed examination of the influence of environmental factors on h₀ using extensive literature data revealed that h₀ may not be constant under certain conditions, particularly at extremes of pH (Delignettemuller, 1998). In fact, it has been shown that pH has a dramatic effect on only when the boundary for growth is reached (Robinson, 1998). These workers observed a marked increase in out of proportion to the effect on at pH values ≤4.8 with the maximum effect at 4.6, whereas our study showed no effect of pH on p₀ to a minimum tested of 4.7. Any attempt to obtain data below a pH of 4.7 resulted in very limited and erratic growth, even when as many as 40–50 replicate wells were examined. In spite of these limitations, it appears likely that p₀ is independent of pH, even at pH values which are close to the limit for growth. Thus, we propose that p₀ is the "true" physiological state parameter, and that pH may have some additional influence on individual cells independent of the physiological state. Elucidation of a novel parameter relating the effect of pH on the lag phase of L. monocytogenes will form the basis of a parallel study (McKellar et al., 2000b).

There was considerable variation in the p₀ values calculated at pH<5.0. As pH values approach the limit for growth, greater variation would be expected; however, our estimate of p₀ based on a large number of Bioscreen experiments gives a reasonably accurate value, assuming that p₀ is, in fact, independent of pH as the data suggest. Other approaches have been taken to model the growth/no growth interface. Ratkowsky and Ross (1995) and others (Bolton; Jenkins and LopezMalo) have used logistic regression to develop probability models for the interface between growth and no growth. It is possible that a kinetic approach may not provide accurate enough estimates of parameters for modelling, particularly when variation in individual cell responses is expected; however, further studies using the Bioscreen should assist in identifying specific parameters describing physiological attributes of individual cells which can be independently tested using such techniques as flow cytometry.

In conclusion, a simple method is proposed which allows the calculation of the p₀ and t_L from t_d data obtained from the Bioscreen. This approach will facilitate studies to determine the influence of environmental factors such as pH and temperature on the behaviour of individual cells.

REFERENCES

Alavi et al., 1999. S.H. Alavi, V.M. Puri, S.J. Knabel, R.H. Mohtar and R.C. Whiting , Development and validation of a dynamic growth model for Listeria monocytogenes in fluid whole milk. Journal of Food Protection 62 (1999), pp. 170-176.

Augustin and Carlier, 2000. J.C. Augustin and V. Carlier , Mathematical modelling of the growth rate and lag time for Listeria monocytogenes. International Journal of Food Microbiology 56 (2000), pp. 29-51.

Baranyi, 1998. J. Baranyi , Comparison of stochastic and deterministic concepts of bacterial lag. Journal of Theoretical Biology 192 (1998), pp. 403-408.

Baranyi and Pin, 1999. J. Baranyi and C. Pin , Estimating bacterial growth parameters by means of detection times. Applied and Environmental Microbiology 65 (1999), pp. 732-736.

Baranyi and Roberts, 1994. J. Baranyi and T.A. Roberts , A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23 (1994), pp. 277-294.

Baranyi et al., 1995. J. Baranyi, T.P. Robinson, A. Kaloti and B.M. Mackey , Predicting growth of Brochothrix thermosphacta at changing temperature. International Journal of Food Microbiology 27 (1995), pp. 61-75.

Bolton and Frank, 1999. L.F. Bolton and J.F. Frank , Defining the growth/no-growth interface for Listeria monocytogenes in Mexican-style cheese based on salt, pH, and moisture content. Journal of Food Protection 62 (1999), pp. 601-609.

Buchanan and Klawitter, 1991. R.L. Buchanan and L.A. Klawitter , Effect of temperature history on the growth of Listeria monocytogenes Scott A at refrigeration temperatures. International Journal of Food Microbiology 12 (1991), pp. 235-246.

Cuppers and Smelt, 1993. H.G.A.M. Cuppers and J.P.P.M. Smelt , Time to turbidity measurement as a tool for modeling spoilage by Lactobacillus. Journal of Industrial Microbiology 12 (1993), pp. 168-171.

Delignettemuller, 1998. M.L. Delignettemuller , Relation between the generation time and the lag time of bacterial growth kinetics. International Journal of Food Microbiology 43 (1998), pp. 97-104.

Delignettemuller et al., 1995. M.L. Delignettemuller, L. Rosso and J.P. Flandrois , Accuracy of microbial growth predictions with square root and polynomial models. International Journal of Food Microbiology 27 (1995), pp. 139-146.

Dufrenne et al., 1997. J. Dufrenne, E. Delfgou, W. Ritmeester and S. Notermans , The effect of previous growth conditions on the lag phase time of some foodborne pathogenic micro-organisms. International Journal of Food Microbiology 34 (1997), pp. 89-94.

Gay et al., 1996. M. Gay, O. Cerf and K.R. Davey , Significance of pre-incubation temperature and inoculum concentration on subsequent growth of Listeria monocytogenes at 14 degrees C. Journal of Applied Bacteriology 81 (1996), pp. 433-438.

Giannuzzi, 1998. L. Giannuzzi , Mathematical modeling of microbial growth in fresh filled pasta stored at different temperatures. Journal of Food Processing and Preservation 22 (1998), pp. 433-447.

Griffiths and Phillips, 1988. M.W. Griffiths and D.M. Phillips , Modeling the relation between bacterial growth and storage temperature in pasteurized milks of varying hygienic quality. Journal of the Society of Dairy Technology 41 (1988), pp. 96-102.

Grijspeerdt and Vanrolleghem, 1999. K. Grijspeerdt and P. Vanrolleghem , Estimating the parameters of the Baranyi model for bacterial growth. Food Microbiology 16 (1999), pp. 593-605.

Hudson, 1993. J.A. Hudson , Effect of pre-incubation temperature on the lag time of Aeromonas hydrophila. Letters in Applied Microbiology 16 (1993), pp. 274-276.

Jenkins et al., 2000. P. Jenkins, P.G. Poulos, M.B. Cole, M.H. Vandeven and J.D. Legan , The boundary for growth of Zygosaccharomyces bailii in acidified products described by models for time to growth and probability of growth. Journal of Food Protection 63 (2000), pp. 222-230.

LopezMalo et al., 2000. A. LopezMalo, S. Guerrero and S.M. Alzamora , Probabilistic modeling of Saccharomyces cerevisiae inhibition under the effects of water activity, pH, and potassium sorbate concentration. Journal of Food Protection 63 (2000), pp. 91-95.

Mackey and Kerridge, 1988. B.M. Mackey and A.L. Kerridge , The effect of incubation temperature and inoculum size on growth of Salmonellae in minced beef. International Journal of Food Microbiology 6 (1988), pp. 57-65.

Martens et al., 1999. D.E. Martens, C. Beal, P.K. Malakar, M.H. Zwietering and K. vantRiet , Modelling the interactions between Lactobacillus curvatus and Enterobacter cloacae I. Individual growth kinetics. International Journal of Food Microbiology 51 (1999), pp. 53-65.

McClure et al., 1993. P.J. McClure, J. Baranyi, E. Boogard, T.M. Kelly and T.A. Roberts , A predictive model for the combined effect of pH, sodium chloride and storage temperature on the growth of Brochothrix thermosphacta. International Journal of Food Microbiology 19 (1993), pp. 161-178.

McKellar, 1997. R.C. McKellar , A heterogeneous population model for the analysis of bacterial growth kinetics. International Journal of Food Microbiology 36 (1997), pp. 179-186.

McKellar and Knight, 2000. R.C. McKellar and K.P. Knight , A combined discrete-continuous model describing the lag phase of Listeria monocytogenes. International Journal of Food Microbiology 54 (2000), pp. 171-180.

McKellar et al., 2000a. R.C. McKellar, X. Lu and K.P. Knight , Development of a dynamic continuous-discrete-continuous model describing the lag phase of individual bacterial cells. In: J.F.M. Van Impe and K. Bernaerts, Editors, Predictive Modelling in Foods-Conference Proceedings, KULeuven/BioTeC, Belgium (2000), pp. 138-140 (ISBN 90-804818-3-1) .

McKellar et al., 2000b. R.C. McKellar, X. Lu and K.P. Knight , Proposal of a novel parameter to describe the influence of pH on the lag phase of Listeria monocytogenes. International Journal of Food Microbiology 73 (2000), pp. 127-135.

McMeekin et al., 1993. T.A. McMeekin, J.N. Olley, T. Ross and D.A. Ratkowsky Predictive Microbiology: Theory and Application, Wiley, New York (1993).

Molina and Giannuzzi, 1999. M. Molina and L. Giannuzzi , Combined effect of temperature and propionic acid concentration on the growth of Aspergillus parasiticus. Food Research International 32 (1999), pp. 677-682.

Ratkowsky and Ross, 1995. D.A. Ratkowsky and T. Ross , Modelling the bacterial growth/no growth interface. Letters in Applied Microbiology 20 (1995), pp. 29-33.

Robinson, 1998. J.A. Robinson , Modeling microbial processes: an overview of statistical considerations. In: A.L. Koch, J.A. Robinson and G.A. Milliken, Editors, Mathematical Modeling in Microbial Ecology, Chapman & Hall, New York (1998), pp. 14-31.

Robinson et al., 1998. T.P. Robinson, M.J. Ocio, A. Kaloti and B.M. Mackey , The effect of the growth environment on the lag phase of Listeria monocytogenes. International Journal of Food Microbiology 44 (1998), pp. 83-92.

Ross and Olley, 1997. T. Ross and J. Olley , Problems and solutions in the application of predictive microbiology. In: F. Shahidi, Y. Jones and D.D. Kitts, Editors, Seafood Safety, Processing, and Biotechnology, Technomic Publishing, Lancaster, PA (1997), pp. 101-118.

Stephens et al., 1997. P.J. Stephens, J.A. Joynson, K.W. Davies, R. Holbrook, H.M. Lappinscott and T.J. Humphrey , The use of an automated growth analyser to measure recovery times of single heat-injured Salmonella cells. Journal of Applied Microbiology 83 (1997), pp. 445-455.

(order Full Text from publisher)

© 2005 Transgalactic Ltd (manufacturer of Bioscreen C software) | Privacy Statement | P.O. Box 1393, 00101 Helsinki, Finland, phone: +358 9 85172920, fax: +358 9 8749481, e-mail: microbiology@bionewsonline.com
 

Last modified: May 25, 2005