Scientific Publications - Work Done by Microbiology Reader Bioscreen C

International Journal of Food Microbiology, 2001, vol. 73, March 11 No.(2-3), pp. 127-135

Proposal of a novel parameter to describe the influence of pH on the lag phase 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

Available online 27 November 2001.


ABSTRACT

Predictive models for lag phase duration () have been less reliable than specific growth rate () models due, in part, to the influence of the pre-growth environment on . A discrete modelling approach was taken to more completely define the response of individual cells to new environments. Time to detection (t_d) data was obtained from serial twofold dilutions of Listeria monocytogenes growing in a Bioscreen at 30 °C. Comparison of the inoculum densities required to achieve maximum t_d at growth pH values from 7.2 to 4.7 revealed that, as the growth pH decreased, fewer cells were capable of making the transition to the exponential phase. The proportion of these cells (termed "adaptable cells") in the original inoculum was used to define a new parameter (r₀) which, when combined with the constant mean individual cell physiological state parameter (p₀), the variation in p₀ (SD_p0), the inital inoculum (N₀) and the maximum population density (N_max) was able to simulate a complete growth curve. Power transformations with rescaled explanatory variables provided suitable models for the influence of pH on , r₀, and SD_p0 (r²>0.70).

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

1. INTRODUCTION

The expression of the growth or death of food-borne microorganisms by mathematical formulations (termed "predictive microbiology") and the development of appropriate delivery systems has been of considerable benefit to the food industry (Buchanan and McClure). Microbial growth can be easily modelled using classical non-linear regression functions such as the Gompertz, which gives good predictions of specific growth rate () but is less effective in predicting population lag phase duration (). Mechanistic models have been proposed in an attempt to more clearly define the physiological changes taking place in the cell during its adaption to new growth environments (Baranyi and Hills); however, these approaches are limited since they consider populations of cells to be homogeneous.

It has been stated that future improvements in growth models must account for the behaviour of individual cells (Baranyi, 1997). McKellar (1997) proposed a model in which the total cell population was dominated by a small growing fraction of the population. Another approach to this problem was suggested by Buchanan et al. (1997) who proposed a model which took into account the variation in adaptation or lag time of individual cells. More recently, Baranyi (1998) described a stochastic model for the lag in which the individual cell lag times were assumed to be identically distributed independent random variables. Baranyi and Pin (1999) have expanded this concept using Bioscreen growth data to estimate growth parameters using an ANOVA procedure.

In our laboratory, we have also used Bioscreen growth data to determine time-to-detection (t_d) values for individual cells of Listeria monocytogenes (McKellar and Knight, 2000). Serial twofold dilutions were used to calculate and individual cell lag times (t_L) and a measure of t_L variability (SD_L) were determined. A discrete-continuous model was then developed in which adaptation of individual cells was represented as discrete events, which when combined with a continuous logistic growth accurately described the transition from lag to exponential phase. This model was further developed by the inclusion of a continuous adaptation phase prior to the discrete step to form a continuous-discrete-continuous (CDC) model, and a new parameter (p₀) was introduced to represent the mean individual cell initial physiological state (McKellar and McKellar). Using this revised model, it was also possible to show that h₀, the population physiological state parameter (Baranyi and McKellar) was equivalent to p₀ for the case of a single cell, but for two or more cells the inter-cell variability must also be considered (Baranyi; McKellar; McKellar and McKellar).

Recent studies on growth of L. monocytogenes in the Bioscreen has demonstrated a simple, geometric method for determining the "true" value of p₀ which is independent of the effects of environmental conditions (McKellar et al., 2000c). It was further demonstrated that p₀ was unaffected by growth pH at close to the limit for growth (pH of 4.7). The present study describes further analysis of the Bioscreen growth data at low pH, and provides support for the proposal of a new lag phase parameter (r₀) which describes the proportion of the inoculum which is capable of growth under a particular set of environmental conditions. A preliminary account of this work was presented previously (McKellar et al., 2000a).

2. MATERIALS AND METHODS

Growth of L. monocytogenes, operation of the Bioscreen, and calculation of t_d and were described previously (McKellar and Knight, 2000). In a parallel study (McKellar and McKellar), we describe the influence of pH on growth of L. monocytogenes in the Bioscreen, and also the method for calculating t_L and p₀. Briefly, serial twofold dilutions of suspensions of L. monocytogenes were made in tryptic soy broth (TSB; Difco Laboratories, Detroit, MI) at pH values ranging from 7.2 to 4.7, and t_d values in the Bioscreen were determined at 30 °C. When t_d values were plotted against inoculum density (ln cfu well⁻¹), straight lines were obtained, the slopes of which could be used to calculate from Eq. (1) (McKellar and Knight, 2000)

Replicate experiments were: 4 for pH 7.2, 6.0 and 4.9; 3 for pH 6.5, 4.8 and 4.7; and 2 for pH 5.5 and 5.0. For each Bioscreen trial, t_L and p₀ were calculated from the x-intercept of the regression of t_d on ln cfu well⁻¹ (McKellar et al., 2000c).

Linear regression and plotting were carried out using Prism™ 3.0 (GraphPad Software for Intuitive Science, San Diego, CA, USA). The basic power model which was fitted to response variables is given in Eq. (2)

where A and B are parameters. This rescaling of the explanatory variable removes local maxima or minima and retains the monotone increasing function (Baranyi and Roberts, 1995). The resulting equations were used with the CDC model (McKellar and McKellar) implemented in ModelMaker© Version 3.0.3 (Cherwell Scientific Oxford, UK) and used to simulate growth curves.

3. RESULTS

Resulting values for were plotted against pH, and the results are shown in Fig. 1. Reducing the pH to 5.5 appeared to have only a minimal effect on , but values decreased markedly below 5.5. Initial attempts to fit a quadratic model to the data were unsuccessful due to the presence of local maxima, thus a rescaling of the explanatory variable (pH) was undertaken. The resulting model with parameters is given in Table 1. A good fit to the model was obtained (r²=0.9611). A similar procedure was adopted for fitting (Fig. 2 and Table 1). A good fit (r²=0.9206) was obtained, and it was again noted that pH had little effect at >5.5. It was also observed that inter-trial variation in and was small, even at low pH (Fig. 1 and Fig. 2).

 

 

Fig. 1. Effect of pH on the specific growth rate () of L. monocytogenes using time to detection (t_d) data obtained from the Bioscreen. Line shows the model fit to the data. Bars represent the SD for replicate trials.

 

Table 1. Models and parameters
 

Fig. 2. Effect of pH on the mean individual cell lag (t_L) of L. monocytogenes using t_d data obtained from the Bioscreen. Line shows the model fit to the data. Bars represent the SD for replicate trials.

 

It had been previously observed (McKellar and McKellar) in plots of t_d against inoculum density (ln cfu well⁻¹) that the cell concentration giving maximum t_d (corresponding to the greatest dilution giving single cells capable of growth) deviated from 0 as the pH decreased. This value (here defined as N_tdmax) is assumed to be equal to 0 when each cell in the inoculum is capable of growth (low stress situations). Changes were relatively slight to as low as pH 4.9; however, at values below that, N_tdmax increased dramatically, and inter-trial variability also increased (Fig. 3).

 

 

Fig. 3. Effect of pH on the inoculum density of the dilution which corresponded to the maximum t_d for each trial (N_tdmax). Line shows a cubic spline. Bars represent the SD for replicate trials.

 

On the assumption that the maximum t_d always corresponds to a single, growing cell, these observations suggest that at low pH, fewer cells initiate growth. This raised the possibility that a new parameter could be proposed which would incorporate into growth models the consequence of a fraction of the inoculum being responsible for growth. The parameter r₀ was therefore defined to express the ratio of a single growing cell to the total inoculum density as in Eq. (3)

The relationship between r₀ and pH is given in Fig. 4. Since r₀=1.0 when 100% of the inoculum was capable of growth, values >1.0 were truncated. The power model with rescaled explanatory variable was appropriate (r²=0.7037) for fitting r₀. It should be noted that, in contrast to N_tdmax (Fig. 3), maximum inter-trial variation was found at pH>5.0, as a result of data transformation.

 

 

Fig. 4. Effect of pH on r₀, the fraction of the total cell population (N₀) which is capable of growth. r₀ values were truncated to 1.0. Line shows the model fit to the data. Bars represent the SD for replicate trials.

 

When a series of binary dilutions was made in the Bioscreen, the variability in t_d between wells increased as the inoculum density approached single cells per well (McKellar and Knight, 2000). This variability was attributed to cell-to-cell variation in t_L, and was defined as the standard deviation of the individual cell lag (SD_L), assuming is constant for each cell (McKellar and Knight, 2000). Further assuming that the variation in t_L can be attributed to inter-cell variability in p₀, it was necessary to scale observed values for SD_L using Eq. (4)

This is based on the relationship between t_L and p₀ as defined previously (McKellar and McKellar) as in Eq. (5)

In an earlier study (McKellar and Knight, 2000), we showed that simulated inter-well variation in SD_L (as calculated using the continuous-discrete [CD] model) decreased as the inoculum level increased in parallel with experimental findings. We are also able to confirm that simulations of SD_L using scaled SD_p0 values in the CDC model also decrease in a similar manner (Fig. 5). Experimental SD_L values for single cells are higher at lower pH, and these decrease in proportion to simulated values (Fig. 5).

 

 

Fig. 5. Comparison of the standard deviation of the t_L (SD_L) from experimental data () and simulated (•) using the continuous-discrete-continuous (CDC) model as influenced by the inoculum density.

 

The relationship between SD_p0 and pH is given in Fig. 6. As the growth pH was reduced, fewer cells were able to initiate growth (smaller r₀), and the variability increased. At pH values <5, it proved difficult to obtain sufficient wells showing growth, even when as many as 50 wells were inoculated. Interestingly, the maximum SD_p0 occurred at pH 4.9, and decreased markedly at lower pH. The inter-trial variability also increased to a maximum at the same point, and decreased at lower pH (Fig. 6). It proved more difficult to fit the power model due to the presence of a discontinuity at pH 4.9. A combined model was employed, with a linear component for pH≤4.9, and a rescaled power model component at pH>4.9. The model showed close agreement (r²=0.7545) with the experimental data (Table 1).

 

 

Fig. 6. Effect of pH on SD_p0, calculated from the variation in t_L among Bioscreen wells for each trial. Line shows the model fit to the data.

 

We are now in a position to expand the CDC model developed earlier (McKellar and McKellar) with functions to describe the influence of pH on , p₀, SD_p0, and r₀. We had previously shown that p₀ was not significantly influenced by pH (McKellar et al., 2000c), and a value of 5.016 derived in that study was use in the CDC model. Parameters for the influence of pH on , SD_p0 and r₀ are given in Table 1. The resulting top-level model as implemented in ModelMaker® is shown in Fig. 7. The mid-level module (the CDCModel sub-model in Fig. 7) which contains the functions for the continuous-discrete-continuous adaptation and growth model has been described previously (McKellar and McKellar). Note that the various blocks in Fig. 7 are of different shape depending on their function: 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; sub-models are rectangular with double edges, and may contain groups of other blocks; and component event blocks are square, and are activated in response to other components in the model.

 

 

Fig. 7. Revised top-level module of the CDC model implemented in ModelMaker®. Description of the various blocks and their function is in the text.

 

The Pre-growth Environment contains a defined value compartment (InitialCount) holding the value for the initial viable cell numbers (N₀), with a component event module (IntialCountTest) to limit model execution to conditions where N₀≤256 cells. A transition section has also been added where the influence of pH is manifested. At t₀, r₀ (defined as a function of Initial_pH) is used to calculate a sub-population of N₀ (AdaptableCells compartment) which limits the total number of cells which are capable of growth. The SD_p0 compartment holds the value of SD_p0 calculated also as a function of pH. As described above, p₀ is independent of pH, and is a defined value.

Simulated output of the revised CDC model is given in Fig. 8 for five different values of pH. N₀ was fixed at 64 cells. It is readily apparent that lowering the pH resulted in a reduced and an extended . Further simulations were performed to visualize the influence of the SD_p0 on the apparent at pH of 7.2, 5.5 and 4.9 (Fig. 9). Five simulations were performed at each pH value with N₀ set to four cells. The number of adaptable cells were calculated by the model as 3, 2 and 1 cells for the 3 pH values, respectively. At lower pH, the increased as seen earlier (Fig. 8); however, in addition the variability among replicate simulations also increased. At pH 4.9, the high SD_p0 was manifested in one of the five simulations giving an apparently shorter than was found at pH 7.2 or 5.5. Different patterns of cell behaviour at pH 4.9 can be obtained by selecting different random number seeds in the simulation (data not shown).

 

 

Fig. 8. Simulated growth curves using the revised CDC. pH values used are: 7.2 (); 6.24 (•); 5.76 (); 5.28 (); and 4.8 (). N₀ was fixed for each pH at 64 cells.

Fig. 9. Results of simulations performed with the revised CDC model with N₀ fixed at four cells. Five replicate simulations were done at each pH with the random number seed changed for each replicate.

4. DISCUSSION

Bacterial cell populations have been traditionally thought of as homogeneous; however, evidence is accumulating to suggest that important properties of cells are best represented by distributions (Kubitschek; Wilhelm; Buchanan; McKellar; Stephens; Baranyi; Schaffner; Whiting and Lambert). These findings have prompted the development of a model which incorporates continuous and discrete cell behaviour to allow single cells to be modelled individually (McKellar; McKellar; McKellar and Wu). The present study expands on earlier work to demonstrate that, when pH is reduced to close to the boundary for growth, a subpopulation of resistant cells is selected upon which the subsequent growth of the population depends. The progressive selection of the subpopulation is represented by a new growth parameter (r₀) which appears to be distinct from the more traditional physiological state parameter (p₀ or h₀) (Baranyi; McKellar and McKellar). It has long been suspected that h₀ is a constant (Baranyi and McKellar); however, consistent variation in the value of h₀ has been attributed to growth conditions (Delignettemuller and Augustin), making it difficult to clearly define h₀ independently of the growth environment. We have recently shown that p₀ was independent of growth pH, down to the practical limits for growth (pH of 4.7) (McKellar et al., 2000c). Thus, we are now able, with the use of two distinct parameters—p₀ and r₀—to model the lag phase in a more mechanistic manner. It is apparent that, as pH is reduced, the increases as a result of slower adaptation and growth, as well as the selection of fewer cells for adaptation.

Our results have also shown increased variability in t_L when cells are exposed to extremes of pH. The inter-well variability (attributed to cell-to-cell variation in p₀) increased markedly when the pH was reduced to 4.9, and then decreased at lower pH values. It is not clear why a maximum variation should be seen at pH 4.9. It is possible that, as the pH is reduced to close to the limit for growth, subpopulations of cells are progressively selected. At pH 4.9, a reasonable percentage of the cells can grow; however, many of the cells are likely close to their pH limit, and variability is high. At pH 4.8 and 4.7, only small populations of resistant cells are selected, with consequently smaller inter-cell variation. These observations are supported by results from the analysis of distributions of cells exposed to low pH. At pH 5.0 and below, it was difficult to clearly establish the type of distribution for cells growing in the Bioscreen due to the small number of wells showing growth; however, microscopic studies revealed a bi-modal distribution at low pH, suggesting the presence of more than one sub-population of cells (Wu et al., unpublished). Support for this concept can also be found in the work of others, who have demonstrated single cell behaviour under extreme conditions. Stephens et al. (1997) examined the effect of heat-injury on Salmonella cells. They showed a broad distribution of lag times at low inoculum levels, and postulated that this was due to the different stages of the cell cycle that the cells were in when they were exposed to heat. Similarly, Lambert et al. (1998), in their study of the effect of disinfectants on Staphylococcus aureus showed that increasing disinfectant concentration resulted in longer subsequent lag times, and suggested that the Bioscreen was useful in quantifying the level of injury. In a more recent study, Augustin et al. (2000) showed that cells of L. monocytogenes which had been starved experienced longer lag times at low innocula, and attributed this to an increase in the variation of individual cells lag time when cells are stressed.

We have found that a power function with rescaling of the explanatory variable to eliminate local maxima or minima was an appropriate model for fitting the parameters of the CDC model. Due to a discontinuity in the SD_p0 data, a linear function was combined with the power function at low pH. These models can be used to produce reasonable simulations of microbial growth; however, data gathering using the Bioscreen is time consuming, so it may not be considered the most appropriate way to derive predictive models. In addition, we noted that there was considerable variation in the values of r₀ at pH>5. This can be attributed to the fact that it is difficult to accurately determine which binary dilution corresponds to single cells per well (McKellar and Knight, 2000). Arbitrary selection of the adjacent dilution as single cells (i.e. ln cfu well⁻¹ difference of 0.69) would have a proportionally greater influence on N_tdmax (and thus r₀) under low stress (neutral pH) with an inoculum level of approximately 1 cell per well than it would if the total inoculum per well was much higher, which occurs at lower pH. Thus, there are limitations to the use of the Bioscreen as a tool to develop applied models. It may be more useful to restrict the use of Bioscreen-derived models to gaining a more complete understanding of the physiological changes taking place during adaptation.

Models developed with the Bioscreen may also be more useful if distribution shape could be taken into account. Preliminary results suggest that, at low pH, the shape of the distribution changes from normal at neutral pH (Wu et al., 2000) to lognormal under acidic conditions (Wu et al., unpublished). Other studies have shown that selection of sub-populations of resistant microorganisms gives rise to skewed distributions (Peleg, 1997). At present, there is no simple method for the continuous modification of distribution shape in either continuous or discrete models, thus, in the present situation, a normal distribution is assumed at all pH values.

The concept of energy diversion has been proposed as an alternative hypothesis to describe the influence of reduced pH on growth of microorganisms (Csonka, 1989). In this approach, the intracellular accumulation of protons at acidic pH puts an additional energy burden on the cells to remove these protons and maintain homeostasis, thus cell yield is reduced (Krist and Lambert). Protons are removed from the cell by the H⁺-ATPase, which is involved in maintaining a trans-membrane potential (see earlier references in Suzuki et al., 2000), and it has been recently shown, using a mutant of Streptococcus mutans deficient in ATPase, that ATPase is involved in generating a stoichiometric electrogenic gradient during the lag phase. Lambert and Stratford (1999) have examined the influence of acidic pH on the lag phase of Saccharomyces cerevisiae. They calculated the activity of ATPase as a function of the intracellular pH, and were able to show that the increased lag experienced due to low pH was due to the time required to pump out protons to achieve an internal pH necessary for growth. The energy diversion approach differs from the CDC model in that it assumes a homogeneous cell population, and does not account for heterogeneity of acid tolerance. In the CDC model, growth at low pH is attributed to a small population of resistance cells—which grow after a proportional to —without accounting for any additional energy expenditure. Further studies are needed to resolve the discrepancies between these two hypotheses.

In the present study, considerable difficulty was experienced in obtaining data on single cell behaviour at low pH. The development of more mechanistic models will depend on our ability to more fully understand and predict cell diversity; however, more powerful tools (such as flow cytometry) will be needed. Other effective approaches to studying the response of microorganisms to limiting growth conditions have been suggested. Logistic regression has been used to combine a probability approach with kinetic models describing the influence of environmental factors to describe the growth boundary, or growth/no growth interface as it is referred to (McMeekin; Ratkowsky and Presser). This approach is still empirical; however, it provides a method for examining the physiological mechanisms which give rise to this growth barrier, or ‘Cole's Cliff’ (McMeekin et al., 2000).

There is no doubt that, as stated by Baranyi (1997), further developments in modelling bacterial growth will require a more complete knowledge of the behaviour of individual cells. The growing importance of this developing field is highlighted by Bridson and Gould (2000), who have introduced the concept of "quantal microbiology". This concept addresses the need to consider bacterial cultures as complex mixtures of subpopulations, and indicates that microbiological outcome of further cultivation is unpredictable. Quantal microbiology is considered to be related to classical microbiology as quantum mechanics is to classical physics. Time will tell if quantal microbiology will have as much impact as quantum mechanics on how we view the sub-microscopic world.

REFERENCES

Augustin et al., 2000. J.C. Augustin, A. BrouillaudDelattre, L. Rosso and V. Carlier , Significance of inoculum size in the lag time of Listeria monocytogenes. Applied and Environmental Microbiology 66 (2000), pp. 1706-1710.

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, 1997. J. Baranyi , Simple is good as long as it is enough. Food Microbiology 14 (1997), pp. 189-192

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 and Roberts, 1995. J. Baranyi and T.A. Roberts , Mathematics of predictive food microbiology. International Journal of Food Microbiology 26 (1995), pp. 199-218.

Bridson and Gould, 2000. E.Y. Bridson and G.W. Gould , Quantal microbiology. Letters in Applied Microbiology 30 (2000), pp. 95-98.

Buchanan, 1993. R.L. Buchanan , Developing and distributing user-friendly application software. Journal of Industrial Microbiology 12 (1993), pp. 251-255.

Buchanan et al., 1997. R.L. Buchanan, R.C. Whiting and W.C. Damert , When is simple good enough: a comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiology 14 (1997), pp. 313-326.

Csonka, 1989. L.N. Csonka , Physiological and genetic responses of bacteria to osmotic stress. Microbiological Reviews 53 (1989), pp. 482-488.

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.

Hills and Wright, 1994. B.P. Hills and K.M. Wright , A new model for bacterial growth in heterogeneous systems. Journal of Theoretical Biology 168 (1994), pp. 31-41.

Krist et al., 1998. K.A. Krist, T. Ross and T.A. McMeekin , Final optical density and growth rate; effects of temperature and NaCl differ from acidity. International Journal of Food Microbiology 43 (1998), pp. 195-203.

Kubitschek, 1966. H.E. Kubitschek , Normal distribution of cell generation rates. Nature 209 (1966), pp. 1039-1040.

Lambert et al., 1998. R.J. Lambert, M.D. Johnston and E.A. Simons , Disinfectant testing: use of the Bioscreen Microbiological Growth Analyser for laboratory biocide screening. Letters in Applied Microbiology 26 (1998), pp. 288-292.

Lambert and Stratford, 1999. R.J. Lambert and M. Stratford , Weak-acid preservatives: modelling microbial inhibition and response.Journal of Applied Microbiology 86 (1999), pp. 157-164.

Lambert and vanderOuderaa, 1999. R.J.W. Lambert and M.L.H. vanderOuderaa , An investigation into the differences between the Bioscreen and the traditional plate count disinfectant test methods. Journal of Applied Microbiology 86 (1999), pp. 689-694.

McClure et al., 1994. P.J. McClure, C.D. Blackburn, M.B. Cole, P.S. Curtis, J.E. Jones, J.D. Legan, I.D. Ogden, M.W. Peck, T.A. Roberts, J.P. Sutherland, S.J. Walker and C.d.W. Blackburn , Modelling the growth, survival and death of microorganisms in foods: the UK food micromodel approach. International Journal of Food Microbiology 23 (1994), pp. 265-275.

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, (2000), pp. 138-140.

McKellar et al., 2000b. R.C. McKellar, X. Lu and K.P. Knight , Growth pH does not affect the initial physiological state parameter (p0) of Listeria monocytogenes. In: J.F.M. Van Impe and K. Bernaerts, Editors, (2000), pp. 37-39.

McKellar et al., 2000c. R.C. McKellar, X. Lu and K.P. Knight , Growth pH does not affect the intial physiological state parameter (p0) of Listeria monocytogenes cells. International Journal of Food Microbiology 73 (2000), pp. 137-146.

McMeekin et al., 2000. T.A. McMeekin, K. Presser, D. Ratkowsky, T. Ross, M. Salter and S. Tienungoon , Quantifying the hurdle concept by modelling the bacterial growth/no growth interface. International Journal of Food Microbiology 55 (2000), pp. 93-98.

Peleg, 1997. M. Peleg , Modeling microbial populations with the original and modified versions of the continuous and discrete logistic equations. CRC Critical Reviews in Food Science and Nutrition 37 (1997), pp. 471-490.

Presser et al., 1998. K.A. Presser, T. Ross and D.A. Ratkowsky , Modelling the growth limits (growth no growth interface) of Escherichia coli as a function of temperature, pH, lactic acid concentration, and water activity. Applied and Environmental Microbiology 64 (1998), pp. 1773-1779.

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.

Schaffner, 1998. D.W. Schaffner , Predictive food microbiology Gedanken experiment: why do microbial growth data require a transformation?. Food Microbiology 15 (1998), pp. 185-189.

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.

Suzuki et al., 2000. T. Suzuki, J. Tagami and N. Hanada , Role of F1F0-ATPase in the growth of Streptococcus mutans GS5. Journal of Applied Microbiology 88 (2000), pp. 555-562.

Whiting and Strobaugh, 1998. R.C. Whiting and T.P. Strobaugh , Expansion of the time-to-turbidity model for proteolytic Clostridium botulinum to include spore numbers. Food Microbiology 15 (1998), pp. 449-453.

Wilhelm et al., 1998. R. Wilhelm, O. Heller, M. Bohland, C. Tomaschewski, I. Klein, P. Klauth, W. Tappe, J. Groeneweg, C.J. Soeder, P. Jansen and W. Meyer , Biometric analysis of physiologically structured pure bacterial cultures recovering from starvation. Canadian Journal of Microbiology 44 (1998), pp. 399-404.

Wu et al., 2000. Y. Wu, M.W. Griffiths and R.C. McKellar , A comparison of the Bioscreen method and microscopy for the determination of lag times of individual cells of Listeria monocytogenes. Letters in Applied Microbiology 30 (2000), pp. 468-472.

(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