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Modelling the combined effect of temperature and pH on the rate coefficient for bacterial growth
ELSEVIER
International Journal of Food Microbiology 23 (1994) 295-303
International Journal of Food Microbiology
Review Paper
Modelling the combined effect of temperature and pH on the rate coefficient for bacterial growth K.R. Davey Department of Chemical Engineering, Universityof Adelaide, Adelaide, S.A. 5005,Australia Received 27 August 1993; revision received 6 December 1993; accepted 25 April 1994
Abstract
An extension of a model of Davey (1989a) has been used to model data of Adams et al. (1991) describing the combined effect of temperature and pH on the rate coefficient for growth of Yersinia enterocolitica in liquid media acidified with four different acidulants: sulphuric, citric, lactic and acetic acids. The model explained between 97.8% and 99.1% of the variance in the results. This very good fit between the predictions and the observed data confirms pH can be incorporated into a model that had previously been applied to the combined effect of temperature and water activity, and to the effect of temperature alone. These findings, taken together with previously reported results, demonstrate the value of the model for predicting the combined effect of two environmental factors on the growth rate. Based on results reported in this paper and previously reported results, a generalised model is proposed for the combined effect of three or more environmental factors, that is amenable to testing. Keywords: Temperature-pH effects; Growth rate; Rate coefficient; Microbial growth; Predictive microbiology; Simulation of bacterial growth; Microbiological process modelling; Predictive microbiological modelling; Linear Arrhenius model
I. Introduction
T h e d e v e l o p m e n t of mathematical models to predict the growth and survival of bacteria has e x p a n d e d greatly in recent years ( A n o n y m o u s , 1994; M c M e e k i n et al., 1993). Models can r e d u c e the n e e d for costly laboratory microbiological analyses, and in the longer term are a pre-requisite for c o m p u t e r control of operations and for process optimisations (Davey, 1992b, 1993a). T h e formulation of models to predict the response of microorganisms has b e e n referred to as predictive microbiology ( M c M e e k i n and Olley, 1986; Baird-Parker and Kilsby, 1987; Gould, 1989; 0168-1605/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0168-1605(94)00050-G
296
K.R. Davey / International Journal of Food Microbiology 23 (1994) 295-303
Roberts, 1990; Ross and McMeekin, 1991; McMeekin et al., 1993). The present author believes this description is too restrictive and that it ignores the necessarily co-operative nature, and likely evolution, of the field (Davey, 1993a). Models that reliably predict the combined effect of environmental factors (e.g. temperature, pH, a w) on microorganisms are of greatest interest. The range and types of these models was categorised by Davey (1989a, 1993b) and more recently, and extensively, by McMeekin et al. (1993). These include the non-linear Arrhenius model of Schoolfield and others (Schoolfield et al., 1981; Broughall et al., 1983; Broughall and Brown, 1984; Vivier et al., 1993), a square-root model (Ratkowsky et al., 1982; Ratkowsky et al., 1983; McMeekin et al., 1987), polynomial model forms (Thayer et al., 1987; Buchanan et al., 1989; Gibson and Roberts, 1989) and linear Arrhenius models (Davey, 1989a; 1991; 1993b). Buchanan and Whiting (1993) have recently suggested classifying models as primary, secondary or tertiary. There has been comparison, and some debate, of models (Davey, 1989b; Adair et al., 1989; Kilsby, 1989; Ratkowsky et al., 1991; McMeekin et al., 1993) and discussion and development and use of appropriate terminology (Davey, 1993b). In a series of papers, the model of Davey (1989a, 1991, 1992c) successfully described growth in different foods with a range of bacterial cell types, both with temperature as the sole environmental factor, and with combined temperature and a w. Advantages of this model include its demonstrated high degree of accuracy and wide application (to published data spanning some 55 years and eight independent researchers), ease of use, and that it is accessible (easily obtained) (Davey, 1989a,b,c; Grau and Vanderlinde, 1992). Based on these successful applications, Davey (1993b) suggested a generalised model form: I n k = Co + ClV, + C2V, 2 + C3V2 + C4V~
(1)
where k is the rate coefficient for growth C 0 , . . . , C a, are coefficients to be estimated, and V1 and V2 are, respectively, environmental factors - including the reciprocal of absolute temperature, a w and, speculatively, pH. Here, this proposition is examined for combined temperature and pH using independent and published data.
2. Materials and methods
For combined temperature and pH the model, from Eq. (1), is therefore given by: I n k = C o + C l / T + C 2 / T 2 + C 3 pH + C 4 pH 2
(2)
Eq. (2) shows a quadratic dependence of the rate coefficient for growth on the influence of pH. This quadratic dependence can be seen in the data reported by Munro (1970) for Escherichia coli (Fig. 1). A quadratic form indicates a maximum value of the rate coefficient for growth. Readers should note that three data points are the minimum that can be modelled by the quadratic form. These data of Fig. 1 are therefore not sufficient to test for
K.R. Da rey / International Journal of Food Microbiology 23 (I 994) 295-303
297
11()~
E E E
•:-_=
80
60
40
2O
I
[
l
i
r
5
6,
7
8
9
p H of m e d i u m during growth
Fig. 1. Effect of pH and temperature on the generation time of E. coli (Munro, 1971).
combined temperature and pH - there being only two temperatures, although the effect of pH is clear. It is interesting to observe an apparent quadratic dependence (symmetry about a mid point) also in the data of Holland et al. (1992), where pH is clearly shown to have a quadratic influence on the maximum specific growth rate (/*max) of Micrococcus sedentarius grown in continuous culture (Fig. 2) and the Kluyveromyces marxianus (a yeast) data of Vivier et al. (1993) (Fig. 3).
0-8
~L
0-6
/
m
o 0-4
o m fi E
/
0-2
/
0--0-~.0
\
/
0
0
0 0-0 6
7
8
9
I0
II
pH
Fig. 2. Effect of pH on the maximum specific growth rate (/.£max) of Micrococcus sedentarius in continuous culture (Holland et al., 1992).
298
K.R. Davey / International Journal of Food Microbiology 23 (1994) 295-303
0.6 [ 0.5 0.4 0.3 0.2 0.1 0.0
0
2
4
6 pH
8
10
2
Fig. 3. Variations in the growth rate of strains of Kluyueromyces marxianus (e IP 513; [] CBS 397) as a function of pH (Vivier et al., 1993).
The criterion for the 'goodness of fit' of the model is the percent variance accounted for (% V) (Davey, 1989a, 1991, 1993b). The % V (Snedecor and Cochran, 1969, pp. 403ff.) is obtained from:
(1-r2)(n% V= 1 -
1)
(n-N-l)
(3)
where r 2 is the multiple regression coefficient, n number of observations in the data set, and N number of terms in the model. For Eq. (2), the number of terms is N = 4 (namely, 1/T, 1/T 2, pH, pH2). % V is seen from Eq. (3) to be a more appropriate test than the multiple regression coefficient where there are few data, or a large number of model coefficients. For n >> N, % V ~ r 2. An alternative measure of the goodness of fit, the mean square error (MSE), proposed by Adair et al. (1989), Little et al. (1992) and Vivier et al. (1993), is given by: MSE=~{
(OBS ~-yRED)2 } r /
(4)
where OBS is observed value of the growth rate, and PRED predicted value. Ratkowsky et al. (1991), however, have criticised the appropriateness of MSE as a test for goodness of fit. They showed, in an analysis of published data, that bacterial growth responses such as generation time and lag time became more variable as their mean magnitude increased. The relationship between the rate coefficient for growth, and the generation time (g) and the rate of growth expressed as divisions per time (G) is: k---
In 2
g
-Gln2
(5)
K.R. Davey / International Journal o f Food Microbiology 23 (1994) 295-303
299
3. Results and discussion
The values of model coefficients (Co . . . . , C 4) obtained from regression analyses (Draper and Smith, 1966, pp. 263ff.) of the graphical data of Adams et al. (1991) for Yersinia enterocolitica serotype 0:3, a microorganism important to chilled foods, are presented in Table 1. Also presented is the number of observations in the data set, the range of temperature and pH, and the % V (and MSE) as a measure of the goodness of fit of the model. A detailed comparison of model predictions of the rate coefficient for growth against observed values at each temperature and p H adjusted with acetic acid are summarised in Table 2. The chemicals, culture media (Tryptone Soya Broth) and the determination of growth over a range of temperature and p H are described in detail by Adams et al. (1991). The same form of the model was applicable to results with all four of the acidulants, by substituting in it the values of the regression coefficients from Table 1. The coefficients of the model were all significant. The rate coefficient for any acidulant is obtained by substituting for the coefficients from Table 1. For example, for the sulphuric acid data the rate model for growth is: In k = - 316 + 1.8149 X 1 0 S / T - 2.7555
1 0 7 / T 2 + 5.77 p H - 0.448 pH 2
X
(6) where k is expressed in day-1. Substitution for T = 285 K and pH = 6.0 gives k = 1.058 day -~. From Eq. (5), this corresponds to a generation time, g = 0.655 day, or, approximately G = 1.53 divisions per day. It is evident, from the magnitude of the values of the % V (and very low MSE) of Table 1 and the comparison of Table 2, that the model provides a high degree of accuracy of prediction against observed data. An important feature to note in Table 1 is that each of the coefficients for the four acidulants has a consistent sign: Co, C 2 and C a have a negative sign, and the remaining two coefficients a positive one. The negative sign on C 2 indicates the extent of curvature in an otherwise simple Arrhenius plot of I n k versus 1 / T . Table 2 illustrates that the value of k increases both with increasing temperature and pH, in each acidulant over the range of experiment. Table 2 shows also that the difference between model predictions and observed values of k is uniform over the entire range of the data set, but with a slight over
Table 1 Fit o f t h e m o d e l f o r t h e r a t e c o e f f i c i e n t *: In k = C O+ C 1 / T Acid
H2SO 4 Citric Lactic Acetic
n
48 39 34 27
CO
-
316 173 135 219
C I x 10
5
1.8149 1.0273 0.7143 1.2397
* T in d e g r e e a b s o l u t e ; k in d a y - 1
C 2 x 10
-
2.7555 1.6084 1.1482 1.9255
7
C3
5.77 2.44 7.51 5.74
+ C2/T2+
C4
-
0.448 0.138 0.556 0.388
C 3 pH + C 4 pH 2
Range
%V (MSE)
T(°C),
pH
1-24, 3-25, 2-24, 3-25,
4.5-6.5 5.0-6.5 5.0-6.5 5.5-6.5
98.6 99.1 97.8 99.4
(0.010) (0.003) (0.005) (0.002)
300
K.R. Darkey/International Journal of Food Microbiology 23 (1994) 295-303
Table 2 Comparison of the observed and predicted rate coefficient for growth pH
5.5
6.0
6.5
Temperature (K)
278.6 280.4 282.2 284 285.7 288.6 293.7 298.1 276.8 278.6 280.4 282.2 284 285.7 288.6 293.7 298.1 275 276.8 278.6 280.4 282.2 284 285.7 288.6 292 296.3
k × 10 2 (day-1) Observed
Predicted
7.84 9.61 14.45 16.00 22.07 32.47 64.10 71.94 10.89 14.45 18.52 25.00 34.84 42.37 65.79 106.38 128.21 10.89 16.00 22.08 29.16 38.46 56.18 75.76 90.09 144.93 185.19
7.49 10.29 13.82 18.12 22.92 32.75 53.98 73.65 10.07 14.2/) 19.51 26.19 34.33 43.44 62.06 102.29 139.56 1/).87 15.72 22.16 30.47 40.88 53.60 67.81 96.88 137.55 194.34
estimation of k at the highest temperature and pH combinations. This was true for e a c h o f t h e t h r e e o t h e r a c i d u l a n t s also. T h e m o d e l n e v e r t h e l e s s is a n a c c u r a t e predictor of k at every temperature-pH combination for each of the acidulants. The first derivative of the quadratic form of the model indicates a maximum r a t e o f g r o w t h at v a l u e s of: T = - 2 C 2 / C ~ a n d p H = - C 3 / 2 C 4. T a b l e 3 s u m marises these values of temperature and pH at which a maximum value of the rate c o e f f i c i e n t f o r g r o w t h is p r e d i c t e d in e a c h o f t h e f o u r a c i d u l a n t s . T h e v a l u e s lie
Table 3 Predicted values of temperature and pH at which the rate coefficient for growth is a maximum Acid
T (K)
pH
H 2SO4 Citric Lactic Acetic
304 313 321 311
6.45 8.84 6.75 7.40
K.R. Davey / International Journal of Food Microbiology 23 (1994) 295-303
301
outside the range of experimental data, and it is not known whether they are accurate predictions. The models cannot be reliably extrapolated outside the range of experimental values. The high degree of accuracy of predictions against observed values may permit limited extrapolations however. No further data in this series of experiments were available for testing, The model has therefore successfully fitted all temperature-pH (and temperature-a w) data that were available. Results confirm the proposition of Davey (1993b) and the potential universality of the model form for predicting the effect of two environmental factors in combination on bacterial growth. The model, in common with various microbiological models, has no theoretical foundation. It is categorised as linear Arrhenius because it was formulated from an Arrhenius form to account for curvature in published data, and in statistical terminology (Davey, 1992a) it is a linear model (i.e. a plot of In k versus 1 / T 2 gives a straight line). In contrast to the reports of other researchers (Broughall et al., 1983; McMeekin et al., 1987), no interpretation is yet offered for the significance of the regression coefficients (C o. . . . . C 4) of the model. The hypothesis that the quadratic term acts as a 'modulator' on the simpler Arrhenius model, however, is being investigated. The model form was used (Davey, Lin and Wood, 1978) for the death rate of C. botulinum and denaturation of vitamins B1, and C (Coker et al., 1993) as affected by combined temperature-pH in a simulation of a continuous steriliser. (It appears to be the first of the predictive models for the combined influence of two environmental factors on bacterial response.) Results presented here for combined effect of temperature and pH, and taken together with earlier findings for the combined effect of temperature and aw, strongly suggest a model for growth of a general, that is universal, form that can be written as an additive sum of environmental factors such that: J In k = C 0 + E ( C 2 i _ I V - [ - C2iVi 2) i:1
(7)
where j environmental factors act in combination to affect the growth rate of microorganisms. The model form of Eq. (7) is amenable to testing. This hypothesis is currently being examined against a large data set generously supplied by R.L. Buchanan (Research Leader, USDA, Philadelphia, PA, USA) and a further supplied by A.M. Gibson (CSIRO Division of Food Processing, Food Research Laboratory, Sydney, N.S.W., Australia).
4. Conclusions The effect of pH on the rate coefficient for growth can be incorporated with temperature in the combined model form of Davey. The model gives a very high degree of goodness of fit to independent data for the combined effect of temperature and pH on the growth of Yersinia enterocolitica in four acidulants: sulphuric, citric, lactic and acetic acid. Based on these results and those reported elsewhere
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K.R. Daeey /International Journal of Food Microbiology 23 (1994) 295-303
f o r c o m b i n e d t e m p e r a t u r e a n d a w, a g e n e r a l i s e d m o d e l f o r p r e d i c t i n g t h e c o m b i n e d e f f e c t o f t h r e e o r m o r e e n v i r o n m e n t a l f a c t o r s is p r o p o s e d .
Acknowledgments For kind permission of the copyright holders to reproduce are acknowledged: A c a d e m i c P r e s s , N e w Y o r k ( F i g . 1), J o u r n a l o f A p p l i e d B a c t e r i o l o g y ( F i g . 2) a n d t h e S o c i e t y o f I n d u s t r i a l M i c r o b i o l o g y (Fig. 3).
References Adair, C., Kilsby, D.C. and Whittall, P.T. (1989) Comparison of the Schoolfield (non-linear Arrhenius) model and the square root model for predicting bacterial growth in foods. Food Microbiol. 6, 7-17. Adams, M.R., Little, C.L. and Easter, M.C. (1991) Modelling the effect of pH, acidulant and temperature on the growth rate of Yersinia enterocolitica. J. Appl. Bacteriol. 71, 65-71. Anonymous (1994) Proc. 1st Int. Conf. Application of Predictive Microbiology and Computer Modeling Techniques to the Food Industry, 12-15 April 1992, Tampa, FL, USA, sponsored by the Society of Industrial Microbiology (in press). Baird-Parker, A.C. and Kilsby, D.C. (1987) Principles of predictive food microbiology. J. Appl. Bacteriol. Symp. Suppl. 43S-49S. Baranyi, J. and Roberts, T.A. (1992) A terminology for models in predictive microbiology - a reply to K.R. Davey. Food Microbiol. 9, 353-356. Broughall, J.M. and Brown, C. (1984) Hazard analysis applied to microbial growth in foods: development and application of three-dimensional models to predict bacterial growth. Food Microbiol. 1, 13-22. Broughall, J.M., Anslow, P.A. and Kilsby, D.C. (1983) Hazard analysis applied to microbial growth in foods: development of mathematical equations describing the effect of water activity. J. Appl. Bacteriol, 55, 101-110. Buchanan, R.L., Stahl, H.G. and Whiting, R.C. (1989) Effects and interactions of temperature, pH, atmosphere, sodium chloride, and sodium nitrite on the growth of Listeria monocytogenes. J. Food Prot. 52, 844-851. Buchanan, R.L. and Whiting, R.C. (1993) A classification of models in predictive microbiology - a reply to K.R. Davey. Food Microbiol. 10, 175-177. Coker, G.L., Davey, K.R. and Kristall, Z. (1993) Modelling the combined effect of pH and temperature on the denaturation of vitamins in a tubular steriliser. In: Asia Pacific Conf. Chemical Engineers CHEMECA '93, 26-29 September, Melbourne, Vic., Australia, Vol. 3, pp. 107-112. Davey, K.R. (1989a) A predictive model for combined temperature and water activity on microbial growth during the growth phase. J. Appl. Bacteriol. 67, 483-488. Davey, K.R. (1989b) Modelling of microbial growth in foods. Comparison of the Schoolfield (non-linear Arrhenius) model and the Square Root model for predicting bacterial growth in foods - a reply to C. Adair et al. Food Microbiol. 6, 302-303. Davey, K.R. (1991) Applicability of the Davey (linear Arrhenius) predictive model to the lag phase of microbial growth. J. Appl. Bacteriol. 70, 253-257. Davey, K.R. (1992a) A terminology for models in predictive microbiology. Food Microbiol. 9, 353-356. Davey, K.R. (1992b) Predictive microbiology and process optimisation. Lett. Appl. Microbiol. 14, 127-128. Davey, K.R. (1993a) Predictive microbiological modelling. In: Proc. 2nd Int. Conf. Modelling and Simulation MS '93, International Association for the Advancement of Modelling and Simulation Techniques in Enterprises, 12-14 July, Melbourne, Vic., Australia, pp. 505-514.
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303
Davey, K.R. (1993b) Linear Arrhenius models for bacterial growth and death and vitamin denaturations. J. Ind. Microbiol. 12, 172-179. Davey, K.R., Lin, S.H. and Wood, D.G. (1978) The effect of pH on continuous high-temperature short-time sterilization of liquid. Am. Inst. Chem. Eng. J. 24, 537-540. Draper, N.R. and Smith, H. (1966) Applied Regression Analysis. Wiley, New York. Gibson, A.M. and Roberts, T.A. (1989) Predicting microbial growth: development of a mathematical model to predict bacterial growth responses. Food Aust. 41, 1075-1079. Gould, G. (1989) Predictive mathematical modelling of microbial growth and survival in foods. Food Sci. Technol. Today 3(2), 89-92. Grau, F.H. and Vanderlinde, P.B. (1992) Occurrence, numbers, and growth of Listeria monocytogenes on some vacuum-packaged processed meats. J. Food Prot. 55, 4-7. Holland, K.T., Marshall, J. and Taylor, D. (1992) The effect of dilution rate and pH on biomass and proteinase production by Micrococcus sedentarius grown in continuous culture. J. Appl. Bacteriol. 72, 429-434. Kilsby, D.C. (1989) Comparison of the Schoolfield (non-linear Arrhenius) model and the square root model for predicting bacterial growth in foods - a reply to K.R. Davey. Food Microbiol. 6, 308-309. Little, C.L., Adams, M.R., Anderson, W.A. and Cole, M.B. (1992) Comparison of a quadratic response surface model and a square root model for predicting the growth rate of Yersinia enterocolitica. Lett. Appl. Microbiol. 15, 63-68. McMeekin, T.A., Chandler, R.E., Doe, P.E., Garland, C.D., Olley, J., Putro, S. and Ratkowsky, D.A. (1987) Model for combined effect of temperature and salt concentration/water activity on the growth rate of Staphylococcus xylosus. J. Appl. Bacteriol. 62, 543-550. McMeekin, T.A. and Olley, J. (1986) Predictive microbiology. Food Technol. Aust. 38, 331-334. McMeekin, T.A., Olley, J.N., Ross, T. and Ratkowsky, D.A. (1993) Predictive Microbiology - Theory and application. Research Studies Press, Taunton, UK. Munro, A.L.S. (1970) Measurement and control of pH values. In: J.R. Norris and D.W. Ribbons (Editors) Methods in Microbiology, 2. Academic Press, New York. Ratkowsky, D.A., Ross, T., McMeekin, T.A. and Olley, J. (1991) Comparison of Arrhenius-type and Bel~hrftdek-type models for prediction of bacterial growth in foods. J. Appl. Bacteriol. 71,452-459. Ratkowsky, D.A., Olley, J., McMeekin, T.A. and Ball, A. (1982) Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149, 1-5. Ratkowsky, D.A., Lowry, R.K., McMeekin, T.A., Stokes, A.N. and Chandler, R.E. (1983) Model for bacterial growth rate throughout the entire bio-kinetic temperature range. J. Bacteriol. 154, 1222-1226. Roberts, T.A. (1990) Predictive modelling of microbial growth. Food Technol. Int., Europe, 231-235. Ross, T. and McMeekin, T.A. (1991) Predictive microbiology - applications of a square root model. Food Aust. 43, 202-207. Schoolfield, R.M., Sharpe, J.P.H. and Magnusson, C.E. (1981) Non-linear regression of biological temperature-dependent rate models based on absolute reaction-rate theory. J. Theor. Biol. 88, 719-731. Snedecor, G.W. and Cochran, W.G. (1969) Statistical Methods, 6th edition. Iowa State University Press, IL, USA. Thayer, D.W., Muller, W., Buchanan, R.L. and Phillips, J.G. (1987) Effect of NaCI, pH, temperature, and atmosphere on growth of Salmonella typhimurium in glucose-mineral salts medium. Appl. Environ. Microbiol. 53, 1311-1315. Vivier, D., Ratomahenina, R., Moulin, G. and Glazy, P. (1993) Study of physiochemical factors limiting the growth of Kluyveromyces marxianus. J. Ind. Microbiol. 11, 157-161.
International Journal of Food Microbiology 23 (1994) 295-303
International Journal of Food Microbiology
Review Paper
Modelling the combined effect of temperature and pH on the rate coefficient for bacterial growth K.R. Davey Department of Chemical Engineering, Universityof Adelaide, Adelaide, S.A. 5005,Australia Received 27 August 1993; revision received 6 December 1993; accepted 25 April 1994
Abstract
An extension of a model of Davey (1989a) has been used to model data of Adams et al. (1991) describing the combined effect of temperature and pH on the rate coefficient for growth of Yersinia enterocolitica in liquid media acidified with four different acidulants: sulphuric, citric, lactic and acetic acids. The model explained between 97.8% and 99.1% of the variance in the results. This very good fit between the predictions and the observed data confirms pH can be incorporated into a model that had previously been applied to the combined effect of temperature and water activity, and to the effect of temperature alone. These findings, taken together with previously reported results, demonstrate the value of the model for predicting the combined effect of two environmental factors on the growth rate. Based on results reported in this paper and previously reported results, a generalised model is proposed for the combined effect of three or more environmental factors, that is amenable to testing. Keywords: Temperature-pH effects; Growth rate; Rate coefficient; Microbial growth; Predictive microbiology; Simulation of bacterial growth; Microbiological process modelling; Predictive microbiological modelling; Linear Arrhenius model
I. Introduction
T h e d e v e l o p m e n t of mathematical models to predict the growth and survival of bacteria has e x p a n d e d greatly in recent years ( A n o n y m o u s , 1994; M c M e e k i n et al., 1993). Models can r e d u c e the n e e d for costly laboratory microbiological analyses, and in the longer term are a pre-requisite for c o m p u t e r control of operations and for process optimisations (Davey, 1992b, 1993a). T h e formulation of models to predict the response of microorganisms has b e e n referred to as predictive microbiology ( M c M e e k i n and Olley, 1986; Baird-Parker and Kilsby, 1987; Gould, 1989; 0168-1605/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0168-1605(94)00050-G
296
K.R. Davey / International Journal of Food Microbiology 23 (1994) 295-303
Roberts, 1990; Ross and McMeekin, 1991; McMeekin et al., 1993). The present author believes this description is too restrictive and that it ignores the necessarily co-operative nature, and likely evolution, of the field (Davey, 1993a). Models that reliably predict the combined effect of environmental factors (e.g. temperature, pH, a w) on microorganisms are of greatest interest. The range and types of these models was categorised by Davey (1989a, 1993b) and more recently, and extensively, by McMeekin et al. (1993). These include the non-linear Arrhenius model of Schoolfield and others (Schoolfield et al., 1981; Broughall et al., 1983; Broughall and Brown, 1984; Vivier et al., 1993), a square-root model (Ratkowsky et al., 1982; Ratkowsky et al., 1983; McMeekin et al., 1987), polynomial model forms (Thayer et al., 1987; Buchanan et al., 1989; Gibson and Roberts, 1989) and linear Arrhenius models (Davey, 1989a; 1991; 1993b). Buchanan and Whiting (1993) have recently suggested classifying models as primary, secondary or tertiary. There has been comparison, and some debate, of models (Davey, 1989b; Adair et al., 1989; Kilsby, 1989; Ratkowsky et al., 1991; McMeekin et al., 1993) and discussion and development and use of appropriate terminology (Davey, 1993b). In a series of papers, the model of Davey (1989a, 1991, 1992c) successfully described growth in different foods with a range of bacterial cell types, both with temperature as the sole environmental factor, and with combined temperature and a w. Advantages of this model include its demonstrated high degree of accuracy and wide application (to published data spanning some 55 years and eight independent researchers), ease of use, and that it is accessible (easily obtained) (Davey, 1989a,b,c; Grau and Vanderlinde, 1992). Based on these successful applications, Davey (1993b) suggested a generalised model form: I n k = Co + ClV, + C2V, 2 + C3V2 + C4V~
(1)
where k is the rate coefficient for growth C 0 , . . . , C a, are coefficients to be estimated, and V1 and V2 are, respectively, environmental factors - including the reciprocal of absolute temperature, a w and, speculatively, pH. Here, this proposition is examined for combined temperature and pH using independent and published data.
2. Materials and methods
For combined temperature and pH the model, from Eq. (1), is therefore given by: I n k = C o + C l / T + C 2 / T 2 + C 3 pH + C 4 pH 2
(2)
Eq. (2) shows a quadratic dependence of the rate coefficient for growth on the influence of pH. This quadratic dependence can be seen in the data reported by Munro (1970) for Escherichia coli (Fig. 1). A quadratic form indicates a maximum value of the rate coefficient for growth. Readers should note that three data points are the minimum that can be modelled by the quadratic form. These data of Fig. 1 are therefore not sufficient to test for
K.R. Da rey / International Journal of Food Microbiology 23 (I 994) 295-303
297
11()~
E E E
•:-_=
80
60
40
2O
I
[
l
i
r
5
6,
7
8
9
p H of m e d i u m during growth
Fig. 1. Effect of pH and temperature on the generation time of E. coli (Munro, 1971).
combined temperature and pH - there being only two temperatures, although the effect of pH is clear. It is interesting to observe an apparent quadratic dependence (symmetry about a mid point) also in the data of Holland et al. (1992), where pH is clearly shown to have a quadratic influence on the maximum specific growth rate (/*max) of Micrococcus sedentarius grown in continuous culture (Fig. 2) and the Kluyveromyces marxianus (a yeast) data of Vivier et al. (1993) (Fig. 3).
0-8
~L
0-6
/
m
o 0-4
o m fi E
/
0-2
/
0--0-~.0
\
/
0
0
0 0-0 6
7
8
9
I0
II
pH
Fig. 2. Effect of pH on the maximum specific growth rate (/.£max) of Micrococcus sedentarius in continuous culture (Holland et al., 1992).
298
K.R. Davey / International Journal of Food Microbiology 23 (1994) 295-303
0.6 [ 0.5 0.4 0.3 0.2 0.1 0.0
0
2
4
6 pH
8
10
2
Fig. 3. Variations in the growth rate of strains of Kluyueromyces marxianus (e IP 513; [] CBS 397) as a function of pH (Vivier et al., 1993).
The criterion for the 'goodness of fit' of the model is the percent variance accounted for (% V) (Davey, 1989a, 1991, 1993b). The % V (Snedecor and Cochran, 1969, pp. 403ff.) is obtained from:
(1-r2)(n% V= 1 -
1)
(n-N-l)
(3)
where r 2 is the multiple regression coefficient, n number of observations in the data set, and N number of terms in the model. For Eq. (2), the number of terms is N = 4 (namely, 1/T, 1/T 2, pH, pH2). % V is seen from Eq. (3) to be a more appropriate test than the multiple regression coefficient where there are few data, or a large number of model coefficients. For n >> N, % V ~ r 2. An alternative measure of the goodness of fit, the mean square error (MSE), proposed by Adair et al. (1989), Little et al. (1992) and Vivier et al. (1993), is given by: MSE=~{
(OBS ~-yRED)2 } r /
(4)
where OBS is observed value of the growth rate, and PRED predicted value. Ratkowsky et al. (1991), however, have criticised the appropriateness of MSE as a test for goodness of fit. They showed, in an analysis of published data, that bacterial growth responses such as generation time and lag time became more variable as their mean magnitude increased. The relationship between the rate coefficient for growth, and the generation time (g) and the rate of growth expressed as divisions per time (G) is: k---
In 2
g
-Gln2
(5)
K.R. Davey / International Journal o f Food Microbiology 23 (1994) 295-303
299
3. Results and discussion
The values of model coefficients (Co . . . . , C 4) obtained from regression analyses (Draper and Smith, 1966, pp. 263ff.) of the graphical data of Adams et al. (1991) for Yersinia enterocolitica serotype 0:3, a microorganism important to chilled foods, are presented in Table 1. Also presented is the number of observations in the data set, the range of temperature and pH, and the % V (and MSE) as a measure of the goodness of fit of the model. A detailed comparison of model predictions of the rate coefficient for growth against observed values at each temperature and p H adjusted with acetic acid are summarised in Table 2. The chemicals, culture media (Tryptone Soya Broth) and the determination of growth over a range of temperature and p H are described in detail by Adams et al. (1991). The same form of the model was applicable to results with all four of the acidulants, by substituting in it the values of the regression coefficients from Table 1. The coefficients of the model were all significant. The rate coefficient for any acidulant is obtained by substituting for the coefficients from Table 1. For example, for the sulphuric acid data the rate model for growth is: In k = - 316 + 1.8149 X 1 0 S / T - 2.7555
1 0 7 / T 2 + 5.77 p H - 0.448 pH 2
X
(6) where k is expressed in day-1. Substitution for T = 285 K and pH = 6.0 gives k = 1.058 day -~. From Eq. (5), this corresponds to a generation time, g = 0.655 day, or, approximately G = 1.53 divisions per day. It is evident, from the magnitude of the values of the % V (and very low MSE) of Table 1 and the comparison of Table 2, that the model provides a high degree of accuracy of prediction against observed data. An important feature to note in Table 1 is that each of the coefficients for the four acidulants has a consistent sign: Co, C 2 and C a have a negative sign, and the remaining two coefficients a positive one. The negative sign on C 2 indicates the extent of curvature in an otherwise simple Arrhenius plot of I n k versus 1 / T . Table 2 illustrates that the value of k increases both with increasing temperature and pH, in each acidulant over the range of experiment. Table 2 shows also that the difference between model predictions and observed values of k is uniform over the entire range of the data set, but with a slight over
Table 1 Fit o f t h e m o d e l f o r t h e r a t e c o e f f i c i e n t *: In k = C O+ C 1 / T Acid
H2SO 4 Citric Lactic Acetic
n
48 39 34 27
CO
-
316 173 135 219
C I x 10
5
1.8149 1.0273 0.7143 1.2397
* T in d e g r e e a b s o l u t e ; k in d a y - 1
C 2 x 10
-
2.7555 1.6084 1.1482 1.9255
7
C3
5.77 2.44 7.51 5.74
+ C2/T2+
C4
-
0.448 0.138 0.556 0.388
C 3 pH + C 4 pH 2
Range
%V (MSE)
T(°C),
pH
1-24, 3-25, 2-24, 3-25,
4.5-6.5 5.0-6.5 5.0-6.5 5.5-6.5
98.6 99.1 97.8 99.4
(0.010) (0.003) (0.005) (0.002)
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K.R. Darkey/International Journal of Food Microbiology 23 (1994) 295-303
Table 2 Comparison of the observed and predicted rate coefficient for growth pH
5.5
6.0
6.5
Temperature (K)
278.6 280.4 282.2 284 285.7 288.6 293.7 298.1 276.8 278.6 280.4 282.2 284 285.7 288.6 293.7 298.1 275 276.8 278.6 280.4 282.2 284 285.7 288.6 292 296.3
k × 10 2 (day-1) Observed
Predicted
7.84 9.61 14.45 16.00 22.07 32.47 64.10 71.94 10.89 14.45 18.52 25.00 34.84 42.37 65.79 106.38 128.21 10.89 16.00 22.08 29.16 38.46 56.18 75.76 90.09 144.93 185.19
7.49 10.29 13.82 18.12 22.92 32.75 53.98 73.65 10.07 14.2/) 19.51 26.19 34.33 43.44 62.06 102.29 139.56 1/).87 15.72 22.16 30.47 40.88 53.60 67.81 96.88 137.55 194.34
estimation of k at the highest temperature and pH combinations. This was true for e a c h o f t h e t h r e e o t h e r a c i d u l a n t s also. T h e m o d e l n e v e r t h e l e s s is a n a c c u r a t e predictor of k at every temperature-pH combination for each of the acidulants. The first derivative of the quadratic form of the model indicates a maximum r a t e o f g r o w t h at v a l u e s of: T = - 2 C 2 / C ~ a n d p H = - C 3 / 2 C 4. T a b l e 3 s u m marises these values of temperature and pH at which a maximum value of the rate c o e f f i c i e n t f o r g r o w t h is p r e d i c t e d in e a c h o f t h e f o u r a c i d u l a n t s . T h e v a l u e s lie
Table 3 Predicted values of temperature and pH at which the rate coefficient for growth is a maximum Acid
T (K)
pH
H 2SO4 Citric Lactic Acetic
304 313 321 311
6.45 8.84 6.75 7.40
K.R. Davey / International Journal of Food Microbiology 23 (1994) 295-303
301
outside the range of experimental data, and it is not known whether they are accurate predictions. The models cannot be reliably extrapolated outside the range of experimental values. The high degree of accuracy of predictions against observed values may permit limited extrapolations however. No further data in this series of experiments were available for testing, The model has therefore successfully fitted all temperature-pH (and temperature-a w) data that were available. Results confirm the proposition of Davey (1993b) and the potential universality of the model form for predicting the effect of two environmental factors in combination on bacterial growth. The model, in common with various microbiological models, has no theoretical foundation. It is categorised as linear Arrhenius because it was formulated from an Arrhenius form to account for curvature in published data, and in statistical terminology (Davey, 1992a) it is a linear model (i.e. a plot of In k versus 1 / T 2 gives a straight line). In contrast to the reports of other researchers (Broughall et al., 1983; McMeekin et al., 1987), no interpretation is yet offered for the significance of the regression coefficients (C o. . . . . C 4) of the model. The hypothesis that the quadratic term acts as a 'modulator' on the simpler Arrhenius model, however, is being investigated. The model form was used (Davey, Lin and Wood, 1978) for the death rate of C. botulinum and denaturation of vitamins B1, and C (Coker et al., 1993) as affected by combined temperature-pH in a simulation of a continuous steriliser. (It appears to be the first of the predictive models for the combined influence of two environmental factors on bacterial response.) Results presented here for combined effect of temperature and pH, and taken together with earlier findings for the combined effect of temperature and aw, strongly suggest a model for growth of a general, that is universal, form that can be written as an additive sum of environmental factors such that: J In k = C 0 + E ( C 2 i _ I V - [ - C2iVi 2) i:1
(7)
where j environmental factors act in combination to affect the growth rate of microorganisms. The model form of Eq. (7) is amenable to testing. This hypothesis is currently being examined against a large data set generously supplied by R.L. Buchanan (Research Leader, USDA, Philadelphia, PA, USA) and a further supplied by A.M. Gibson (CSIRO Division of Food Processing, Food Research Laboratory, Sydney, N.S.W., Australia).
4. Conclusions The effect of pH on the rate coefficient for growth can be incorporated with temperature in the combined model form of Davey. The model gives a very high degree of goodness of fit to independent data for the combined effect of temperature and pH on the growth of Yersinia enterocolitica in four acidulants: sulphuric, citric, lactic and acetic acid. Based on these results and those reported elsewhere
302
K.R. Daeey /International Journal of Food Microbiology 23 (1994) 295-303
f o r c o m b i n e d t e m p e r a t u r e a n d a w, a g e n e r a l i s e d m o d e l f o r p r e d i c t i n g t h e c o m b i n e d e f f e c t o f t h r e e o r m o r e e n v i r o n m e n t a l f a c t o r s is p r o p o s e d .
Acknowledgments For kind permission of the copyright holders to reproduce are acknowledged: A c a d e m i c P r e s s , N e w Y o r k ( F i g . 1), J o u r n a l o f A p p l i e d B a c t e r i o l o g y ( F i g . 2) a n d t h e S o c i e t y o f I n d u s t r i a l M i c r o b i o l o g y (Fig. 3).
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Davey, K.R. (1993b) Linear Arrhenius models for bacterial growth and death and vitamin denaturations. J. Ind. Microbiol. 12, 172-179. Davey, K.R., Lin, S.H. and Wood, D.G. (1978) The effect of pH on continuous high-temperature short-time sterilization of liquid. Am. Inst. Chem. Eng. J. 24, 537-540. Draper, N.R. and Smith, H. (1966) Applied Regression Analysis. Wiley, New York. Gibson, A.M. and Roberts, T.A. (1989) Predicting microbial growth: development of a mathematical model to predict bacterial growth responses. Food Aust. 41, 1075-1079. Gould, G. (1989) Predictive mathematical modelling of microbial growth and survival in foods. Food Sci. Technol. Today 3(2), 89-92. Grau, F.H. and Vanderlinde, P.B. (1992) Occurrence, numbers, and growth of Listeria monocytogenes on some vacuum-packaged processed meats. J. Food Prot. 55, 4-7. Holland, K.T., Marshall, J. and Taylor, D. (1992) The effect of dilution rate and pH on biomass and proteinase production by Micrococcus sedentarius grown in continuous culture. J. Appl. Bacteriol. 72, 429-434. Kilsby, D.C. (1989) Comparison of the Schoolfield (non-linear Arrhenius) model and the square root model for predicting bacterial growth in foods - a reply to K.R. Davey. Food Microbiol. 6, 308-309. Little, C.L., Adams, M.R., Anderson, W.A. and Cole, M.B. (1992) Comparison of a quadratic response surface model and a square root model for predicting the growth rate of Yersinia enterocolitica. Lett. Appl. Microbiol. 15, 63-68. McMeekin, T.A., Chandler, R.E., Doe, P.E., Garland, C.D., Olley, J., Putro, S. and Ratkowsky, D.A. (1987) Model for combined effect of temperature and salt concentration/water activity on the growth rate of Staphylococcus xylosus. J. Appl. Bacteriol. 62, 543-550. McMeekin, T.A. and Olley, J. (1986) Predictive microbiology. Food Technol. Aust. 38, 331-334. McMeekin, T.A., Olley, J.N., Ross, T. and Ratkowsky, D.A. (1993) Predictive Microbiology - Theory and application. Research Studies Press, Taunton, UK. Munro, A.L.S. (1970) Measurement and control of pH values. In: J.R. Norris and D.W. Ribbons (Editors) Methods in Microbiology, 2. Academic Press, New York. Ratkowsky, D.A., Ross, T., McMeekin, T.A. and Olley, J. (1991) Comparison of Arrhenius-type and Bel~hrftdek-type models for prediction of bacterial growth in foods. J. Appl. Bacteriol. 71,452-459. Ratkowsky, D.A., Olley, J., McMeekin, T.A. and Ball, A. (1982) Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149, 1-5. Ratkowsky, D.A., Lowry, R.K., McMeekin, T.A., Stokes, A.N. and Chandler, R.E. (1983) Model for bacterial growth rate throughout the entire bio-kinetic temperature range. J. Bacteriol. 154, 1222-1226. Roberts, T.A. (1990) Predictive modelling of microbial growth. Food Technol. Int., Europe, 231-235. Ross, T. and McMeekin, T.A. (1991) Predictive microbiology - applications of a square root model. Food Aust. 43, 202-207. Schoolfield, R.M., Sharpe, J.P.H. and Magnusson, C.E. (1981) Non-linear regression of biological temperature-dependent rate models based on absolute reaction-rate theory. J. Theor. Biol. 88, 719-731. Snedecor, G.W. and Cochran, W.G. (1969) Statistical Methods, 6th edition. Iowa State University Press, IL, USA. Thayer, D.W., Muller, W., Buchanan, R.L. and Phillips, J.G. (1987) Effect of NaCI, pH, temperature, and atmosphere on growth of Salmonella typhimurium in glucose-mineral salts medium. Appl. Environ. Microbiol. 53, 1311-1315. Vivier, D., Ratomahenina, R., Moulin, G. and Glazy, P. (1993) Study of physiochemical factors limiting the growth of Kluyveromyces marxianus. J. Ind. Microbiol. 11, 157-161.