Mathematical modelling of the combined effect of water activity, pH and redox potential on the heat destruction
O l i v 6 r R e i c h a r t a,* C s i l l a M o h f i c s i - F a r k a s b
a University of Horticulture and Food Industry. Department of Microbiology and Biotechnology, Somldi fit 14-16, Budapest H-1118, Hungary
b Research Institute for Canning Industry, F6ldvdry u. 4, Budapest H-1097, Hungary
Abstract
Heat destruction of seven foodborne microorganisms (Lactobacillus plantarum, Lacto- bacillus brevis, Saccharomyces cerevisiae, Zygosaccharomyces bailii, Yarrowia lipolytica, Pae- cilomyces varioti and Neosartoria fischeri) as a function of the temperature, pH, redox potential and water activity was studied in synthetic heating media. Several mathematical models were developed for describing the heat destruction rate, most of them resulted in a good correlation between the fitted and measured values. The determination coefficients of the model-fitting were the best in case of lactobacilli and moulds (0.96-0.99) and the worst in case of the yeasts (0.81-0.88).
Keywords: Heat destruction; pH; Redox potential; Water activity; Mathematical modelling
1. Introduct ion
In descr ib ing the t h e r m a l or chemica l des t ruc t ion of microorganisms , the most commonly used m a t h e m a t i c a l m o d e l is b a s e d on the analogy of the f i r s t -o rde r react ions , i.e.
d N - - - k N (1) d t
whe re N is the concen t r a t i on of the living cells, t is the t ime and k is the specific d e a t h - r a t e coeff ic ient . Eq. (1) was p r o p o s e d by Chick (1908) and is genera l ly e m p l o y e d for the k ine t ic analysis of mic rob ia l des t ruc t ion processes .
104 O. Reichart, C. Moh dcsi-Farkas / International Journal of Food Microbiology 24 (1994) 103-112
To calculate the number of the surviving cells is possible by solving the differential equation (1). The only problem is the value of the death-rate coeffi- cient (k), because it depends on the temperature, disinfectant concentration, pH, water activity, redox potential and other environmental factors.
The temperature dependence of the rate-coefficient may be expressed in several ways (Arrhenius, 1889; Bigelow, 1921; Eyring, 1935a,b).
Since the publication of the transition state theory (Eyring, 1935a), the Eyring's model is the most widely used conceptual scheme for discussing reaction rates beside the Arrhenius equation, and is frequently applied in describing the thermal properties of microorganisms in original (Van Uden, 1984; Moser, 1988) and in modified form, as it is reviewed by Ratkowsky et al. (1991).
Investigation of the effect of the water activity on the heat resistance resulted in the following conclusions. The heat resistance increases with the decreasing water activity (Goepfert et al., 1970; Corry, 1974; Verrips and Kwast, 1977; Verrips et al., 1979). Studying the thermal inactivation of sugar-tolerant yeasts, T6r6k and Reichart (1983) established a negative linear relationship between the logarithm of the apparent activation energy and the sucrose concentration of the heating menstruum. These results are in conformity with the model suggested by Moser (1988):
- EA(aw) ( 2 ) k = ko(aw)exp RT
where k are rate-coefficients, and ko and E A are the constants of the Arrhenius equation, depending on the water activity (aw).
Jermini and Schmidt-Lorenz (1987) investigated the heat resistance of vegeta- tive cells and asci of osmotolerant yeasts in two different broths of a w 0.963 and 0.858, respectively. They demonstrated, that asci of Zygosaccharomyces bailii had a D-value at 60°C and a w 0.858 of 14.9 rain. The heat resistance of asci (represented by the D-values) at a w 0.963 proved to be 5- to 8-fold higher than those of the vegetative cells. However, the lower the a w of the heating broth, the smaller the differences between the heat resistance of asci and vegetative cells.
The heat resistance of Neosartorya fischeri isolated from commercial fruit juice was studied by Scott and Bernard (1987). Under certain conditions this mould had a D-value of 1.4 rain at 87°C and z-value of 5.6°C, so it could survive the commercial thermal process if ascospores were present in sufficient numbers.
Studying the effect of pH on the thermal death time in the range of pH 2.8-8.8, Jordan and Jacobs (1948) obtained a linear relationship between the logarithm of thermal death time and pH in the acid and alkaline range. These results are in conformity with the theoretical interpretation of the effect of pH on the heat destruction (Reichart, 1994): the logarithm of the heat destruction rate increases linearly in the acid and alkaline range and has a minimum at the optimum pH of the growth.
Although redox potential is believed to be an important selective factor for the growth of microbes, there are not available relevant references on the effect of redox potential on the heat resistance. Even the mathematical models of the
O. Reichart, C. Mohdcsi-Farkas /International Journal of Food Microbiology 24 (1994) 103-112 105
predictive microbiology, summarized by McMeekin et al. (1993), do not include the redox potential.
This paper deals with the mathematical modelling of the combined effect of environmental factors on the heat resistance.
Thermal destruction of seven foodborne microorganisms as a function of the temperature, water activity, pH and redox potential was studied in synthetic media in the acid range of soft drinks and pasteurized foods.
2. Materials and methods
2.1. Microbial strains
Microbial strains used in the experiments were obtained from the National Collection of Agricultural and Industrial Microorganisms (Budapest, Hungary): Lactobacillus brevis, Lactobacillus plantarum, Saccharomyces cerevisiae, Zygosac- charomyces bailii, Yarrowia lipolytica, Paecilomyces varioti, Neosartorya fischeri.
2.2. Culture media
For culturing yeasts a medium containing 1.0% (w/w) glucose, 1.0% (w/w) peptone, 0.3% (w/w) yeast extract and 1.5% (w/w) agar was used (made from OXOID components). Lactobacilli were cultured on MRS-agar (OXOID CM 359) plates. Moulds were cultured on YEA plates containing 2.0% (w/w) glucose, 0.5 % (w/w) yeast extract and 1.5% (w/w) agar. (Made from OXOID components.)
2.3. Heating media
Different amounts of glycerol were added to 0.5% (w/w) glucose solution in order to control water activity (Handbook of Chemistry and Physics, 1985-1986). The water activity used in the experiments were 0.850, 0.920, 0.940, 0.960, 0.980 and 0.995. The glycerol concentrations were 42.0, 27.5, 22.5, 16.5, 9.0 and 2.5% (m/m), respectively.
The redox potential was controlled by Na-thioglycolate (SIGMA) according to Fields (1979, pp. 85-86). The pH was adjusted with HC1 and NaOH solutions.
The pH and redox potential of the heating media were measured with a digital mV- and pH-meter (RADELKIS OP-201, Budapest, Hungary). The redox poten- tial was measured with Pt and calomel electrodes and the Eh-values were calcu- lated according to Jacob (1970):
E h = E m + E c ( 3 )
where E h is the electrode potential referred to the normal hydrogen electrode, E m is the measured potential and E c is the potential of the reference (calomel) electrode.
106 o. Reichart, C. Mohdcsi-Farkas /International Journal of Food Microbiology 24 (1994) 103-112
The rH of the heating media was calculated from the E h and pH-values (Webb, 1964):
rH = 2 ( E h + 0.1984 T . p H ) / ( O . 1 9 8 4 T ) (4)
where E h is substituted in mV terms, and T is the absolute temperature. The environmental factors of the heat treatments were as follows. The rH-val-
ues were calculated from pH and E h according to Eq. (4).
Lactobacillus brevis Zygosaccharomyces bailii T (°C) 51, 54, 57 T (°C) 52, 56, 60 a w 0.92, 0.94, 0.96, 0.98 a w 0.850, 0.920, 0.995 pH 3.5, 4.0, 4.5 pH 3.0, 3.5, 4.0, 4.5 E h (mV) 150, 240, 460 a E h (mV) 150, 240, 460 a rH 13-23 rH 13-23
Lactobacillus plantarum Paecilomyces varioti T (°C) 55, 58, 61 T (°C) 60, 65, 70 a w 0.92, 0.94, 0.96, 0.98 a w 0.850, 0.920 0.995 pH 3.5, 4.0, 4.5 pH 3.0, 3.5, 4.0, 4.5 E h (mV) 150, 240, 460 a E h (mV) 150, 240, 460 a rH 13-23 b rH 13-23
Saccharomyces cerevisiae Neosartorya fischeri T (°C) 52, 56, 60 T (°C) 80, 85, 90 a w 0.850, 0.920, 0.995 a w 0.850, 0.920, 0.995 pH 3.0, 3.5, 4.0, 4.5 pH 3.0, 3.5, 4.0, 4.5 E h (mV) 110, 240, 460 a E h (mV) 150, 240, 460 a rH 13-23 b rH 13-23
Yarrowia lipolytica T (°C) 48, 50, 52 a w 0.850, 0.920, 0.995 pH 3.0, 3.5, 4.0, 4.5 E h (mV) 110, 240, 460 ~ rH 12-23
a The values were varying in the treatments. b Calculated from pH and E h.
2.4. Heat treatments
Lactobacillus strains and yeasts. After 48 h of incubation, four agar slopes were washed down with 10 cm 3 heating menstruum at room temperature. The cells were centrifuged at 2500 g for 10 rain, then the pellet was resuspended in 5 cm 3 heating medium at room temperature. The suspension was added to 200 cm 3 of preheated and thermostated heating medium, completely stirred by a magnetic stirrer.
Moulds. After 30 days incubation at 30°C, the agar plates were washed down with the heating medium and the Neosartorya fischeri suspensions were heated at 70°C
O. Reichart, C. Mohdcsi-Farkas / International Journal of Food Microbiology 24 (1994) 103-112 107
T a b l e 1 R e g r e s s i o n coef f i c i en t s o f t he m a t h e m a t i c a l m o d e l s o f t he h e a t d e s t r u c t i o n o f Lactobacillus brevis
M o d e l Coe f f i c i en t s fo r t h e m o d e l p a r a m e t e r s
p a r a m e t e r s M o d e l
1 2 3 4 5
a - 18.56 - 19.21 61.01 - 10.83 68 .74
p H - 0 .1383 - - 0 .1384 - 0 .1384 - 0 .1384
E b ( m V ) 0 .000424 - 0 .000424 0 .000424 0 .000424
r H - 0 .01116 - - -
a w 7.738 7 .759 7 .739 - - lg a w - - - 16.929 16.929
T = 5 1 - 5 7 ° C ; p H = 3 . 5 - 4 . 5 ; E b = 1 1 0 - 4 8 0 mV; r H = 1 3 - 2 3 ; a w = 0 . 9 2 0 - 0 . 9 8 0 ; n u m b e r o f d a t a - p a i r s =
108.
f o r 1 0 m i n t o k i l l t h e v e g e t a t i v e c e l l s . T h e i n o c u l u m l e v e l s w e r e a b o u t 1 0 7 c f u / c m 3,
d e t e r m i n e d b y p l a t e c o u n t i n g .
Heating system. T h e h e a t t r e a t m e n t e x p e r i m e n t s w e r e r u n i n o p e n f l a s k o f 3 0 0 c m 3
v o l u m e c o n t a i n i n g 2 0 0 c m 3 h e a t i n g m e d i u m . T h e h e a t i n g m e n s t r u u m w a s s t i r r e d
w i t h T e f l o n - c o a t e d m a g n e t i c s t i r r e r a n d t h e t e m p e r a t u r e w a s c o n t r o l l e d b y u l t r a -
t h e r m o s t a t e d w a t e r , c i r c u l a t i n g i n a s u b m e r g e d g l a s s h e a t i n g s p i r a l .
T a b l e 2
R e g r e s s i o n coe f f i c i en t s o f t he m a t h e m a t i c a l m o d e l s o f t h e h e a t d e s t r u c t i o n o f Lactobacillus plantarum
M o d e l Coe f f i c i en t s fo r t he m o d e l p a r a m e t e r s
p a r a m e t e r s M o d e l
1 2 3 4 5
a - 17.07 - 18.15 48.19 - 8.46 56.80
p H - 0 .2466 - - 0 .2465 - 0 .2466 - 0 .2465
E b ( m V ) 0 .000608 - 0 .000609 0 .000608 0 .000609 r H - 0 .0146 - -
a w 8.620 8.618 8 .620 - -
lg aw - - - 18.87 18.87 T (°C) 0 .1678 0 .1679 - 0 .1678 -
1 /Tab s ( K - l) _ - - 18380 - - 18380
R 2 0 .968 0.911 0 ,967 0 .968 0 .968
z (°C) 5.96 5.96 - 5.96 -
T = 5 5 - 6 1 ° C ; p H = 3 . 5 - 4 . 5 ; E h = 1 1 0 - 4 6 0 mV; r H = 1 3 - 2 3 ; aw = 0 . 9 2 0 - 0 . 9 8 0 ; n u m b e r o f d a t a - p a i r s = 108.
108 O. Reichart, C. Mohdcsi-Farkas / International Journal of Food Microbiology 24 (1994) 103-112
At appropriate intervals, 1.0 cm 3 sample was taken from the heated and stirred suspension and immediately cooled down by pipetting it into dilution medium (0.1% bacto peptone in physiological NaCI solution, pH = 6.8).
The number of viable cells was determined by plate counting in the substrates mentioned above. The thermal death-rate coefficient ( k ) w a s calculated from the regression equation of the survival curve.
2.5. Mathemat ica l model,;
The mathematical models of the heat destruction, proposed by this paper are based on the kinetic model of Reichart (1994), and extend the Arrhenius (1889) and Bigelow (1921) models in order to describe the effect of the pH, water activity and redox potential on the thermal destruction rate.
The applied mathematical models are: 1. log m k = a l + b l . p H + b 2 E h + b 3 a w + b a T
2. log10 k = a 2 + b 5 . rH + b3a w + b4T 3. log m k = a 3 + bj • pH + b 2 E h + b3a w + b 6 / T a b s
4. loglo k = a 4 + b l ' p H + b z E h + b 7 lg a w + b 4 T 5. loglo k = a 5 + b l - p H + b 2 E h + b 7 lg a w + b 6 / T a b s The coefficients of the model-parameters a~_ 5 and b l _ 7 w e r e calculated with multiparametric linear regression, using the STATGRAPHICS 5.1 program pack- age (Statistical Graphics Corporation, USA).
3. Results and discussion
For the calculation of the coefficients of models 1-5, multiparametric linear regression was applied, where the dependent variables (lg k) were obtained by log10 transformation of the thermal death rate (k) of rain -~ units.
T a b l e 3
R e g r e s s i o n c o e f f i c i e n t s o f t he m a t h e m a t i c a l m o d e l s o f t he h e a t d e s t r u c t i o n o f Saccharomyces cereuisiae
M o d e l C o e f f i c i e n t s fo r t he m o d e l p a r a m e t e r s
p a r a m e t e r s M o d e l
1 2 3 4 5
a - 9.588 - 10.56 57.97 - 8.063 59.46
p H - 0.4031 - - 0.4031 - 0 , 4 0 3 l - 0.4031
E h ( m Y ) 0 .00188 - 0 .00188 0 ,00188 0 .00188
r H 0.0613 -
a w 1.499 1.519 1.499 -
lg a,,. - 3 .135 3.135
T (°C) 0.1755 0 .1752 - 0 .1755
1 /Tab ~ ( K - I ) - - - - 18982 - - 18982
R 2 0.845 0.795 0.845 0.845 0 .844
z (°C) 4.79 4.79 - 4.79 -
T = 5 2 - 6 0 ° C ; p H = 3 . 0 - 4 . 5 ; E h = 1 2 0 - 5 1 0 m V ; r H = 1 3 - 2 3 , a w = 0 . 8 5 0 - 0 . 9 9 5 ; n u m b e r o f d a t a - p a i r s =
108.
O. Reichart, C. Mohdcsi-Farkas / International Journal of Food Microbiology 24 (1994) 103-112 109
T a b l e 4
Regression coefficients of the m a t h e m a t i c a l m o d e l s o f the heat destruction o f Yarrowia lipolytica
M o d e l C o e f f i c i e n t s f o r t h e m o d e l p a r a m e t e r s
T = 4 8 - 5 2 ° C ; p H = 3 . 0 - 4 . 5 , E h = 1 4 0 - 5 1 0 m V ; r H = 1 3 - 2 3 , a w = 0 . 8 5 0 - 0 . 9 9 5 ; n u m b e r o f d a t a - p a i r s =
108.
The results of the model-fittings are summarized in Tables 1-7, where the z-values (calculated as z = 1 / b 4 from models 1, 2 and 4) are depicted as well.
As shown in Tables 1-7, model-fitting is the worst in the case of model 2, containing the rH parameters. Applying the other models, there are no significant differences between the determination coefficients (R2). The best fitting can be obtained by using model 1.
Comparing the fitting of the heat destruction models the correlation proved to be the best in the case of lactobacilli ( R 2 = 0.96-0.99) and the worst in case of yeasts (R 2 = 0.81-0.88).
T a b l e 5
Regression coefficients of the m a t h e m a t i c a l m o d e l s o f the heat destruction o f Zygosaccharomyces bailii
M o d e l C o e f f i c i e n t s f o r t h e m o d e l p a r a m e t e r s
T = 5 2 - 6 0 ° C ; p H = 3 . 0 - 4 . 5 ; E n = 7 0 - 5 1 0 m V ; r H = 1 3 - 2 3 , a w = 0 . 8 5 0 - 0 . 9 9 5 ; n u m b e r o f d a t a - p a i r s =
108.
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T a b l e 6 R e g r e s s i o n c o e f f i c i e n t s o f t h e m a t h e m a t i c a l m o d e l s o f t h e h e a t d e s t r u c t i o n o f Neosartorya fischeri
M o d e l C o e f f i c i e n t s f o r t h e m o d e l p a r a m e t e r s
T = 8 0 - 9 0 ° C ; p H = 3 . 0 - 4 . 5 ; E h = 1 4 0 - 5 1 0 m V ; r H = 1 3 - 2 3 ; aw = 0 . 8 5 0 - 0 . 9 9 5 ; n u m b e r o f d a t a - p a i r s =
108.
Taking into account the individual regression coefficients of pH, E h, a,~, T and l / T a b ~, the mathematical models resulted in the same values and determination coefficients for the microorganism tested (see the horizontal rows in the tables), so for the prediction of the heat destruction rate model equations 1, 3, 4 and 5 can be applied equally.
For simple modelling, equations 1 and 4 could be recommended; for reaction kinetic study, equations 3 and 5.
T a b l e 7
R e g r e s s i o n c o e f f i c i e n t s o f t h e m a t h e m a t i c a l m o d e l s o f t h e h e a t d e s t r u c t i o n o f Paecilomyces varioti
M o d e l C o e f f i c i e n t s f o r t h e m o d e l p a r a m e t e r s
T = 6 0 - 7 0 ° C ; p H = 3 . 0 - 4 . 5 ; E h = 1 4 0 - 5 3 0 m V ; r H = 1 3 - 2 3 , a~, = 0 . 8 5 0 - 0 . 9 9 5 ; n u m b e r o f d a t a - p a i r s =
108.
O. Reichart, C. Mohdcsi-Farkas / International Journal of Food Microbiology 24 (1994) 103-112 111
T h e effect of p H and wa te r activity on the hea t des t ruc t ion ( r ep resen ted by the
sign of the regress ion coeff icients) was similar at every microorganism: the hea t
des t ruc t ion ra te has increased with the decreas ing p H and increasing wate r
activity. The effect of the redox po ten t ia l d e p e n d e d on the kind of microorganisms.
Increas ing the redox po ten t ia l of the hea t ing med ium, the heat des t ruc t ion of
lactobacil l i acce lera ted , while the hea t resis tance of yeasts and moulds increased.
Acknowledgements
This work was suppor ted by the Nat iona l Commiss ion for Technolog ica l Devel -
o p m e n t of Hunga ry (OMFB) , and represen ts a par t of the research p rogram in the
growth, survival and hea t inact ivat ion of food-spoi lage microorganisms.
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