Death rates of bacterial spores: mathematical models1
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YOUNW . H A N ,HWE ~ IK Z H A N GA, N D J O H NM . KROCHTA Agric~rrltrirrrlRcsrrrrclr Srr~nice,U.S. Depcirtt?~etl/~fAgr.icriltrrrc,,Brrhrley. Col(fort~ici94710 nt1d Deprrrttnrtlt u/'Pl~ysic.s,Sroril Nntiotrnl Ut~i~.e,sity, Scorrl, Korerr Accepted October 14, 1975 H A N ,Y . W.. H. I. ZHANC,and J . M . KROCHTA.1976. Death ratesofbacterial spores: mathematical models. Can. J. Microbiol. 22: 295-300. The concave survivor curves produced a s a result of spore heterogeneity were analyzed to determine whether they were caused by innate characteristicsof the spores or by theacquisition of heat resistance during the heating process. Mathematical models developed for the two hypotheses revealed that the concave survivor curve (on semi-log paper) caused by innate heterogeneity is parabolic and that caused by acquired heat resistance is exponential. The mathematical models were applied to several published survivor curves of different organisms, and heat resistance parameters and the cause of curviline:lrity were determined. For the cases r innate studied, the cause of curvilinearity appears to be acquisition of heat resistance ~ x t h e than heterogeneity of spore population.
H A N ,Y.W.. H. I. Z H A N C ~ M. ~ J KROCHTA. . 1976. Death ratesof bacterial spores: mathematical models. Can. J . Microbiol. 22: 295-300. Les courbes concaves de survivance obtenues par suite d e I'heterogeneite des spores furent analyskes pour determinersi elles resultent de caracttres innes ou de I'acquisition parces spores d'une resistance h la chaleur au cours d e traitements 5 la chaleur. Des modtles mathematiques developpCs pour ces deux hypotheses revelent que la courbe de I'heterogCneite innee e s t parabolique et celle qui resulte d e la resistance h la chaleur ncquise est exponentielle. C e s modeles mathematiques f~irentappliques i divers organismes, dont les c o ~ i r b e sde survivance s e retrouvent en litterature, et les parametres de resistance a la chaleur ainsi que la cause d e curvilinearite furent determinis. Pour les cas etudies. la cause de curvilinCarite parait liee h I'acquisition d e resistance h la chaleur plut6t qu'a une heterogeneitt innee d e In population d e s spores. [Traduit par le journal]
Introduction The death rate of bacteria and their spores subjected t o lethal agents is generally considered t o be logarithmic. However, in many instances bacterial death rate is not logarithmic. The cause of nonlinear survivor curves has been explained by the multiple critical sites theory (Johnson et 01.1954; Meynell and Meynell 1970; Moats 1971; Rahn 1943; Wood 1956), experimental artifacts (Stumbo 1965), and heterogeneity of spore heat resistance (Han 1975). Heterogeneity, however, can be visualized in two ways: ( I ) the heat resistance of spore population is heterogeneous before any heat treatment (innate heterogeneity theory); (2) the heat resistance of spore population is initially homogeneous, but the heterogeneity is developed dur'Received August 18, 1975. *Present address: ARS, U S D A , Department of Microbiology, Oregon State University, - ~ o r v a l l i s , Oregon 97331.
ing the heating process as a result of heat adaptation by the spores (heat adaptation theory (Alderton et ul. 1964; Komemushi et 01. 1968)). Both theories result in similar survivor curves. This paper reports the ~nathernaticalmodels for the concave survivor curves produced by innate heterogeneity and by heat adaptation of bacterial spores.
Mathematical Models and Discussion Two mathematical models, based o n ( I ) innate heterogeneity of spore population and (2) heat adaptation during a heating period, were developed. F o r mathematical description, "rate of destruction," K, is defined as the probability of destruction of a spore per unit time. This value is assumed t o be a constant for the first model (innate heterogeneity model) and to be a function of time, t , for the second model (heat adaptation model). K is related t o the heat resistance in such a way that those spores with a larger K have smaller resistance.
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296
CAN. J. MICROBIOL. VOL. 22, 1976
M(rtlien7atical Mo(/cl for Innate Heterogeneity Tl~eo,y Tlie initial distribution of the destruction rate of a given population is determined by the following consideration. The given population is assumed to be heterogeneous with regard to the heat resistance. Then the resistance distribution would be purely statistical. Because the distribution pattern of bacterial lieat resistance is not known, it is assumed to be a binomial distribution, which reduces to the normal distribution (Gaussian distribution) as the number of samples (spores) beconies large enough. However, the Gaussian distribution allows nonzero population having extreme values in both directions. This situation is unrealistic and we need to truncate at some reasonable limits. The truncated normal distribution of the initial rate of destruction is tlien
CII
c(t,K) = c(0,K) exp (-Kt) 131 From Eqs. 1 and 3
x exp
[-
((K - 5 ) ' 20
x exp
+~ t ) ]
(- ~ . t +t2 2 1 "'
The total distribution C(t) is given by
1 p exp (- (K
~ ( o , K )= ~ ( 0 )
for ( K - KoI < a c(0,K)
=
- (KO- 02t)] 2/202)dK
x exp (- [K
0 for IK - KoI > a
where c(0,K) is the concentration of spores with rate of destruction K at time 0, KOis the most probable value of K among the spores, o is the standard deviation, and a is the truncation parameter. The standard deviation o represents the degree of heterogeneity for a given population of spores. It is essential for the following discussion toe~timatethe order of magnitude of o and a. We therefore assume that o = 0.1 KO, a = 0.5 KO.For these values of o and a,
and C(0)
& =S
J:zo
exp
($)
dx
x [exp (- Kot + 02t2/2)] Let ~ ( x= )
c(0,K) dK
KO-n
This means that the constant C(0) is the total concentration at time 0. The time development of c(t,K) satisfies
exp (- t2/2) dt
J-xm
tlien the change of variable [K - (KO- 02t)]/o > ill the above integralshows that
,
+ +
C(t) = C(O)[exp (-Kot 02t2/2)] x [P(ot a/o) - P(at - a/o)] so that
+ log [P(ot + a/o) - P(ot - a/o)]
=1
K,fa
+
The value ofP(x) ateacli xis tabulated in standard matliematical tables (Abramowitz and Segun 1965). For the estimated values of a and o, and for t = 20/K0, which is large enough (20 Do) to cover most experimental ranges,
[ (o r + -3- (
P ot--
log P
91
=-0.0013
For the other terms, which is integrated to give
-Kot = -20,
o2 -t2 2
=4
HAN E T AL.: D E A T H RATES O F BACTERIAL SPORES
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We can therefore omit for time t 5 20/K0 the term log [P(ot + ala) - P ( o t - crlo)] in Eq. 5 and get
Equation 5 represents a parabolic curve. Therefore it is concluded that tlie innate heterogeneity of spore population produces parabolic survivor curves on semi-log paper during at least the initial 20 deci~ualreduction time (20 Do), whicli is beyond any experimentally measurable time.
297
of destructive power. Assumption (ii) can be written as
where B is another constant representing the rate of development of resistance. Equation 9 lnay be integrated to give
From this equation, the constant B can be interpreted as the ti~nerequired to develop the resistance, R,,,,,[I - exp (- I)], which is about Mathonerticcrl Moclelfor Heat Adcrptcrtioi~T11eory 639, of the maximum attainable value. The following were assumed for the forniulaFrom Eqs. 7 , 8 , and 10, tion o f a matliematical model for the heat adaptation theory: (1) spores develop resistance when [l I] K(t) = KO{] - cc[l - exp (- tlB)]) exposed to lieat; (2) there exists a maximum level of resistance attainable by each spore before it which gives the time dependency of rate of dedies; (3) it takes some finite time to acquire a struction in ternis of two parameters a and B. substantial amount of resistance. The rate of Substituting for K(t) in Eq. 6, a differential destruction, K , is defined to be a function of heat- equation satisfied by the spore concentration, ing time, t, for tlie heat adaptation model. There- C(t), is obtained: fore the destruction rate, K(t), satisfies the following equation, Ko{l - cc[1 - exp ( - t/B)])C(t)
where C(t) is tlie concentration of spores. If K(t) is restricted to time independency, this equation reduces to the ordinary logarithmic order of death rate. The resistance, R(t), i n the 'Pores at a particular time, t, is defined as the amount of decrease i n the rate of destruction from the initial rate of destruction, KO,namely,
This equation can easily be soJved:
Now, we assume further that, (i) the maximum level of resistance that can be developed a t a given temperature is proportional to the initial rate of destruction, which represents the severity of destructive power, and (ii) the rate of increase in resistance is proportional to the difference between the maximum resistance and the degree of resistance already developed. Assumption (i) can be expressed as
Pcrrameters of Hecrt Resistance of Bcrcteriul Spores The proposed heat adaptation model was applied to two different species ofbacterial spores, Bcrcillus cougulrms strain 43P and a strain of Bacillus cereus. Their thermal destruction rates were studied by Frank and Campbell (1957) and Han et a/. (1971), respectively. By their data, the values of a , B, and Do were determined by a leastsquares fit of Eq. 13. The values obtained are given in Table I . Here Do is defined t o be ]/KO. The computed survival curves, obtained by use of these parameters, are presented in Figs. 1 and 2 with the corresponding experimental values.
[8I Rmax = aKo where cc is a constant representing the maximum amount of resistance attainable for a unit amount
[I 31 log C(t)
=
log Co - KO - ctB[exp(--t/B) - I])
x {(I - a)t
where Co is the initial concentration of spores. Thus t h e curvilinear survivor curve caused by heat adaptation is exponential in nature. Equation 13 can be readily reduced to logarithmic order if the constant a vanishes.
C A N . J. MICROBIOL. VOL. 22, 1976
TABLE1. Thermal resistance parameters of B, coagulans and B. cereus
Do,min
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B. coagulansa B. cereusc
a
B, min
80°C
0.66 0.87
0.31 4.2
85°C
90°C
95°C
-
-
9.03
2.82
1.52
1.24
-
-
107.2"C 0.431b -
-
nThermal-destruction data obtained from Fig. 1 of Frank and Campbell (1957). bFor the survival curve of lowest initial spore concentration, Dowas found to be 0.362. Thermal-destruction data obtained from Fig. I of Han et al. (1971).
TlME IMin)
F I G . I . Survivor curves for spores of B. cereus heated at different temperatures. ( a ) Experimental values; (-) calculated curve. Thermal-resistance data obtained from Fig. 1 of Han e/ 01. (1971).
T h e a values for B. coagrilai~sand B. cereus were calculated to be 0.66 and 0.87, respectively. They do not necessarily indicate that the spores of B. caeus develop more resistance to heat than d o those of B. conguln17s, but the relative resistance, in terms of Rm,,/K,, developed in B. cerelts is larger than that in B. coogulans. Theoretically, organisms having a values larger than 1 could not be killed by heat, because the resistance developed in the spore during a finite time, R ( t ) , exceeds the destruction power, KO.Also, we can speculate tliat the straight-line survivor curve results from innate homogeneity and inability to develop resistance ( a = 0) in the organism for some reason, such as heating a t high temperature and for too short a time. In B. congulnris and B. cereus, the B values, which represent the rate of developnient of resistance, were 0.31 and 4.2, respectively. They indicate tliat spores of B. coag~tlcir7sdevelop resistance faster than those of B. cereus. This rate of development niay be an innate characteristic of each organism or a factor depending o n the environment. the rate of heat For spores of B. 11zegcrteri~o17, adaptation was temperature dependent (Alderton et (11. 1964). If this dependence is valid for B. coctg~tk(~ns and B. cereus, then the B values merely reflect the environment, rather than innate characteristics of the organism. T h e difference in B values between B. cong~ilar~s and B. cereus (Table 1) can then be explained as a result of difference in temperature used. At a higher temperature, shorter time is required for development of heat resistance. Appliccrtiori of the Mollels
The proposed mathen~aticalmodels dimerentiate the nature of the curves caused by innate TlME ( M i n ) heterogeneity and heat adaptation of the spores. F I G . 2. Survivor curves for spores of B. coogulot~s The former yields a parabolic curve and the latter heated at 107.2'C. ( a ) Experimental values; (-) calcuyields an exponential curve on semi-log paper, as lated curves. Thermal-resistance data obtained from s l ~ o w nin Figs. 3 and 4. As mentioned earlier, the Fig. 1 of Frank and Campbell (1957).
HAN E T AL.: DEATH RATES O F BACTERIAL SPORES
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LOG
%
c\
d LOG
(7,
dt
TIME
-
FIG. 5. Differential of innate-heterogeneity model, Eq. 5. T h e result 1s a s t r a ~ g h tline.
TIME-
FIG. 3. Survivor curve resulting from innate heterogeneity, showing parabolic nature. Curve applies only for KO 2 oZt.
d LOG
(&)
TIME-
FIG. 6. Differential of heat adaptation model, Eq. 13. The res~rltis an initial curvature tending t o horizontal straight line. I
TIME
-
1
FIG. 4. Survivor curve resulting from heat adaptation, showing exponential nature.
parabolic curve derived from the innate heterogeneity theory applied at least for t 5 20/K0, that is, the time required to reduce 20 initial log cycles (20 Do). These two curves have features worth noting. Both have slopes of -KO as t approaches zero. The curve resulting from the heat adaptation theory approaches a straight line at a sufficiently large time. This line has a slope of -Ko(l - ci) and intercepts the ordinate at a value of -aBKo. Thus, if accurate data are available at sufficiently small and large times, a , B a n d K Ocan all be determined graphically.
Quite often, however, data are not adequate to differentiate between the two models by plotting on semi-log paper. Thus, it is difficult t o distinguish visually whether a concave curve is parabolic o r exponential. Whether the experimental survivor curve is parabolic or exponential can be distinguished by comparison of the slope of the survivor curve, dldt log C(t)/Co, as a function of time. Whereas the differential of the parabolic curve (Eq. 5) produces a straight line with slope a', as shown in Fig. 5, the differential of the exponential curve (Eq. 13) produces the curve shown in Fig. 6. This approach has been applied to the data of Figs. 1 and 2 with results appearing in Fig. 7. Data points are indicated; the curves result from use of the appropriate parameter values sub-
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C A N . J. MICROBI(3L. VOL. 22. 1976
dt
-14
0 DATA
OF CURVE
@,
0 DATA
OF CURVE
@ , Fng 2
Fse 2
TIME (MIN)
FIG. 7. (A) Differential of 8 0 ° C and 95 "C data o f Fig. 1. (B) Differential of data from curves 1 and 2 of Fig. 2. Curvature confirms that heat adaptation model applies.
stituted in Eq. 13. Data at 85 "C and 90 "C from Fig. 1 give equally good results, but are not included in Fig. 7 for clarity. Data from curve 3 of Fig. 2 were insufficient t o test. Results in Fig. 7 clearly show that the cause of curvilinearity in the survival curves of B. coagulans and B. cereus is the development of heat resistance by the spores during the heating period. In the natural case, where both heterogeneity and heat adaptation simultaneously cause the curvature, a con~promisedcurve results, showing both influences. Usually, one effect is more pronounced than the other. Thus, with the vegetative cells, whose population is believed t o be more heterogeneous than that of spores, survivor curves resemble parabolic more than exponential curves, whereas with the spores, the opposite results. Because heat adaptation by the spores is more likely to occur at lower than a t higher tempera-
tures within the lethal range, the survivor curves at different temperatures also reveal different shapes, showing those more parabolic at higher temperatures but more exponential at lower ones. The mathematical niodels that we have proposed are mainly phenon~enological, and the validity of these models certainly rests o n the experimental data on the destructive mode of bacterial spores. However, once the validity of these models has been established through further experimental evidence, the models will provide a useful way t o describe the thermal-destruction characteristics of each species of bacterium a n d bacterial spores. ABRAMOWITZ, M., and L. A. S E G U N1965. . Handbook of mathematical functions. Dover Publications Inc., New York. N.Y. G., P. A. THOMPSON, and N. S N E L L1964. . ALDERTON, Heat adaptation and ion exchange in Brrci1lrr.s r,ic8gritr.r.i~rttispores. Science, 143: 141-143. F R A N KH. . A,, and L . L. C A M P B E LJLR. . 1957. The nonlogarilhmic rate of thermal destruction of spores of B. corrgrrlrrr~.s.Appl. Microbiol. 5 : 243-248. H A N .Y. W.. H. A. SCHUYTEN. JR., and C. D. C A L L I H A N . 1971. The combined effect of heat and alkali insterilizing sugarcane bagasse. J. Food Sci. 36: 335-338. H A N .Y. W. 1975. Death rates of bacterial spores: nonlinear survivor curves. Can. J . Microbiol. 21: 1464-1467. JOHNSON. F . , H. E Y R I N Gand , M. J. POLISSAR. 1954. The kinetic basis of molecular biology. John Wiley & Sons, Inc., New York. KOMEMUSH S..I . K. OKUBO, and G . T E R U I1968. . Kinetics on the thermal death of microrganisms. IV. On the change of death rate constant of bacterial spores in the course of heat sterilization. J . Ferment. Technol. 46(3): 249-256. M E Y N E L LG., G., and E. M E Y N E L L1970. . Theory and practice in experimental bacteriology. Cambridge University Press. London. MOATS,W. A. 1971. Kinetics of thermal death of bacteria. J . Bacteriol. 105: 165-171. RAHN,0 . 1943. The problems of the logarithmic order of death in bacteria. Biodynamica, 4: 81. STUMBO. C. R. 1965. Thermobacteriology in food processing. Academic Press, lnc.. New York. WOOD.T. H. 1956. Lethal effects of high and low temperatureson ~ i n i c e l l ~ ~organisms. lar Adv. Biol. Med. Phys. 4: 119-165.
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YOUNW . H A N ,HWE ~ IK Z H A N GA, N D J O H NM . KROCHTA Agric~rrltrirrrlRcsrrrrclr Srr~nice,U.S. Depcirtt?~etl/~fAgr.icriltrrrc,,Brrhrley. Col(fort~ici94710 nt1d Deprrrttnrtlt u/'Pl~ysic.s,Sroril Nntiotrnl Ut~i~.e,sity, Scorrl, Korerr Accepted October 14, 1975 H A N ,Y . W.. H. I. ZHANC,and J . M . KROCHTA.1976. Death ratesofbacterial spores: mathematical models. Can. J. Microbiol. 22: 295-300. The concave survivor curves produced a s a result of spore heterogeneity were analyzed to determine whether they were caused by innate characteristicsof the spores or by theacquisition of heat resistance during the heating process. Mathematical models developed for the two hypotheses revealed that the concave survivor curve (on semi-log paper) caused by innate heterogeneity is parabolic and that caused by acquired heat resistance is exponential. The mathematical models were applied to several published survivor curves of different organisms, and heat resistance parameters and the cause of curviline:lrity were determined. For the cases r innate studied, the cause of curvilinearity appears to be acquisition of heat resistance ~ x t h e than heterogeneity of spore population.
H A N ,Y.W.. H. I. Z H A N C ~ M. ~ J KROCHTA. . 1976. Death ratesof bacterial spores: mathematical models. Can. J . Microbiol. 22: 295-300. Les courbes concaves de survivance obtenues par suite d e I'heterogeneite des spores furent analyskes pour determinersi elles resultent de caracttres innes ou de I'acquisition parces spores d'une resistance h la chaleur au cours d e traitements 5 la chaleur. Des modtles mathematiques developpCs pour ces deux hypotheses revelent que la courbe de I'heterogCneite innee e s t parabolique et celle qui resulte d e la resistance h la chaleur ncquise est exponentielle. C e s modeles mathematiques f~irentappliques i divers organismes, dont les c o ~ i r b e sde survivance s e retrouvent en litterature, et les parametres de resistance a la chaleur ainsi que la cause d e curvilinearite furent determinis. Pour les cas etudies. la cause de curvilinCarite parait liee h I'acquisition d e resistance h la chaleur plut6t qu'a une heterogeneitt innee d e In population d e s spores. [Traduit par le journal]
Introduction The death rate of bacteria and their spores subjected t o lethal agents is generally considered t o be logarithmic. However, in many instances bacterial death rate is not logarithmic. The cause of nonlinear survivor curves has been explained by the multiple critical sites theory (Johnson et 01.1954; Meynell and Meynell 1970; Moats 1971; Rahn 1943; Wood 1956), experimental artifacts (Stumbo 1965), and heterogeneity of spore heat resistance (Han 1975). Heterogeneity, however, can be visualized in two ways: ( I ) the heat resistance of spore population is heterogeneous before any heat treatment (innate heterogeneity theory); (2) the heat resistance of spore population is initially homogeneous, but the heterogeneity is developed dur'Received August 18, 1975. *Present address: ARS, U S D A , Department of Microbiology, Oregon State University, - ~ o r v a l l i s , Oregon 97331.
ing the heating process as a result of heat adaptation by the spores (heat adaptation theory (Alderton et ul. 1964; Komemushi et 01. 1968)). Both theories result in similar survivor curves. This paper reports the ~nathernaticalmodels for the concave survivor curves produced by innate heterogeneity and by heat adaptation of bacterial spores.
Mathematical Models and Discussion Two mathematical models, based o n ( I ) innate heterogeneity of spore population and (2) heat adaptation during a heating period, were developed. F o r mathematical description, "rate of destruction," K, is defined as the probability of destruction of a spore per unit time. This value is assumed t o be a constant for the first model (innate heterogeneity model) and to be a function of time, t , for the second model (heat adaptation model). K is related t o the heat resistance in such a way that those spores with a larger K have smaller resistance.
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296
CAN. J. MICROBIOL. VOL. 22, 1976
M(rtlien7atical Mo(/cl for Innate Heterogeneity Tl~eo,y Tlie initial distribution of the destruction rate of a given population is determined by the following consideration. The given population is assumed to be heterogeneous with regard to the heat resistance. Then the resistance distribution would be purely statistical. Because the distribution pattern of bacterial lieat resistance is not known, it is assumed to be a binomial distribution, which reduces to the normal distribution (Gaussian distribution) as the number of samples (spores) beconies large enough. However, the Gaussian distribution allows nonzero population having extreme values in both directions. This situation is unrealistic and we need to truncate at some reasonable limits. The truncated normal distribution of the initial rate of destruction is tlien
CII
c(t,K) = c(0,K) exp (-Kt) 131 From Eqs. 1 and 3
x exp
[-
((K - 5 ) ' 20
x exp
+~ t ) ]
(- ~ . t +t2 2 1 "'
The total distribution C(t) is given by
1 p exp (- (K
~ ( o , K )= ~ ( 0 )
for ( K - KoI < a c(0,K)
=
- (KO- 02t)] 2/202)dK
x exp (- [K
0 for IK - KoI > a
where c(0,K) is the concentration of spores with rate of destruction K at time 0, KOis the most probable value of K among the spores, o is the standard deviation, and a is the truncation parameter. The standard deviation o represents the degree of heterogeneity for a given population of spores. It is essential for the following discussion toe~timatethe order of magnitude of o and a. We therefore assume that o = 0.1 KO, a = 0.5 KO.For these values of o and a,
and C(0)
& =S
J:zo
exp
($)
dx
x [exp (- Kot + 02t2/2)] Let ~ ( x= )
c(0,K) dK
KO-n
This means that the constant C(0) is the total concentration at time 0. The time development of c(t,K) satisfies
exp (- t2/2) dt
J-xm
tlien the change of variable [K - (KO- 02t)]/o > ill the above integralshows that
,
+ +
C(t) = C(O)[exp (-Kot 02t2/2)] x [P(ot a/o) - P(at - a/o)] so that
+ log [P(ot + a/o) - P(ot - a/o)]
=1
K,fa
+
The value ofP(x) ateacli xis tabulated in standard matliematical tables (Abramowitz and Segun 1965). For the estimated values of a and o, and for t = 20/K0, which is large enough (20 Do) to cover most experimental ranges,
[ (o r + -3- (
P ot--
log P
91
=-0.0013
For the other terms, which is integrated to give
-Kot = -20,
o2 -t2 2
=4
HAN E T AL.: D E A T H RATES O F BACTERIAL SPORES
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We can therefore omit for time t 5 20/K0 the term log [P(ot + ala) - P ( o t - crlo)] in Eq. 5 and get
Equation 5 represents a parabolic curve. Therefore it is concluded that tlie innate heterogeneity of spore population produces parabolic survivor curves on semi-log paper during at least the initial 20 deci~ualreduction time (20 Do), whicli is beyond any experimentally measurable time.
297
of destructive power. Assumption (ii) can be written as
where B is another constant representing the rate of development of resistance. Equation 9 lnay be integrated to give
From this equation, the constant B can be interpreted as the ti~nerequired to develop the resistance, R,,,,,[I - exp (- I)], which is about Mathonerticcrl Moclelfor Heat Adcrptcrtioi~T11eory 639, of the maximum attainable value. The following were assumed for the forniulaFrom Eqs. 7 , 8 , and 10, tion o f a matliematical model for the heat adaptation theory: (1) spores develop resistance when [l I] K(t) = KO{] - cc[l - exp (- tlB)]) exposed to lieat; (2) there exists a maximum level of resistance attainable by each spore before it which gives the time dependency of rate of dedies; (3) it takes some finite time to acquire a struction in ternis of two parameters a and B. substantial amount of resistance. The rate of Substituting for K(t) in Eq. 6, a differential destruction, K , is defined to be a function of heat- equation satisfied by the spore concentration, ing time, t, for tlie heat adaptation model. There- C(t), is obtained: fore the destruction rate, K(t), satisfies the following equation, Ko{l - cc[1 - exp ( - t/B)])C(t)
where C(t) is tlie concentration of spores. If K(t) is restricted to time independency, this equation reduces to the ordinary logarithmic order of death rate. The resistance, R(t), i n the 'Pores at a particular time, t, is defined as the amount of decrease i n the rate of destruction from the initial rate of destruction, KO,namely,
This equation can easily be soJved:
Now, we assume further that, (i) the maximum level of resistance that can be developed a t a given temperature is proportional to the initial rate of destruction, which represents the severity of destructive power, and (ii) the rate of increase in resistance is proportional to the difference between the maximum resistance and the degree of resistance already developed. Assumption (i) can be expressed as
Pcrrameters of Hecrt Resistance of Bcrcteriul Spores The proposed heat adaptation model was applied to two different species ofbacterial spores, Bcrcillus cougulrms strain 43P and a strain of Bacillus cereus. Their thermal destruction rates were studied by Frank and Campbell (1957) and Han et a/. (1971), respectively. By their data, the values of a , B, and Do were determined by a leastsquares fit of Eq. 13. The values obtained are given in Table I . Here Do is defined t o be ]/KO. The computed survival curves, obtained by use of these parameters, are presented in Figs. 1 and 2 with the corresponding experimental values.
[8I Rmax = aKo where cc is a constant representing the maximum amount of resistance attainable for a unit amount
[I 31 log C(t)
=
log Co - KO - ctB[exp(--t/B) - I])
x {(I - a)t
where Co is the initial concentration of spores. Thus t h e curvilinear survivor curve caused by heat adaptation is exponential in nature. Equation 13 can be readily reduced to logarithmic order if the constant a vanishes.
C A N . J. MICROBIOL. VOL. 22, 1976
TABLE1. Thermal resistance parameters of B, coagulans and B. cereus
Do,min
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B. coagulansa B. cereusc
a
B, min
80°C
0.66 0.87
0.31 4.2
85°C
90°C
95°C
-
-
9.03
2.82
1.52
1.24
-
-
107.2"C 0.431b -
-
nThermal-destruction data obtained from Fig. 1 of Frank and Campbell (1957). bFor the survival curve of lowest initial spore concentration, Dowas found to be 0.362. Thermal-destruction data obtained from Fig. I of Han et al. (1971).
TlME IMin)
F I G . I . Survivor curves for spores of B. cereus heated at different temperatures. ( a ) Experimental values; (-) calculated curve. Thermal-resistance data obtained from Fig. 1 of Han e/ 01. (1971).
T h e a values for B. coagrilai~sand B. cereus were calculated to be 0.66 and 0.87, respectively. They do not necessarily indicate that the spores of B. caeus develop more resistance to heat than d o those of B. conguln17s, but the relative resistance, in terms of Rm,,/K,, developed in B. cerelts is larger than that in B. coogulans. Theoretically, organisms having a values larger than 1 could not be killed by heat, because the resistance developed in the spore during a finite time, R ( t ) , exceeds the destruction power, KO.Also, we can speculate tliat the straight-line survivor curve results from innate homogeneity and inability to develop resistance ( a = 0) in the organism for some reason, such as heating a t high temperature and for too short a time. In B. congulnris and B. cereus, the B values, which represent the rate of developnient of resistance, were 0.31 and 4.2, respectively. They indicate tliat spores of B. coag~tlcir7sdevelop resistance faster than those of B. cereus. This rate of development niay be an innate characteristic of each organism or a factor depending o n the environment. the rate of heat For spores of B. 11zegcrteri~o17, adaptation was temperature dependent (Alderton et (11. 1964). If this dependence is valid for B. coctg~tk(~ns and B. cereus, then the B values merely reflect the environment, rather than innate characteristics of the organism. T h e difference in B values between B. cong~ilar~s and B. cereus (Table 1) can then be explained as a result of difference in temperature used. At a higher temperature, shorter time is required for development of heat resistance. Appliccrtiori of the Mollels
The proposed mathen~aticalmodels dimerentiate the nature of the curves caused by innate TlME ( M i n ) heterogeneity and heat adaptation of the spores. F I G . 2. Survivor curves for spores of B. coogulot~s The former yields a parabolic curve and the latter heated at 107.2'C. ( a ) Experimental values; (-) calcuyields an exponential curve on semi-log paper, as lated curves. Thermal-resistance data obtained from s l ~ o w nin Figs. 3 and 4. As mentioned earlier, the Fig. 1 of Frank and Campbell (1957).
HAN E T AL.: DEATH RATES O F BACTERIAL SPORES
Can. J. Microbiol. Downloaded from www.nrcresearchpress.com by UNIV WINDSOR on 11/15/14 For personal use only.
LOG
%
c\
d LOG
(7,
dt
TIME
-
FIG. 5. Differential of innate-heterogeneity model, Eq. 5. T h e result 1s a s t r a ~ g h tline.
TIME-
FIG. 3. Survivor curve resulting from innate heterogeneity, showing parabolic nature. Curve applies only for KO 2 oZt.
d LOG
(&)
TIME-
FIG. 6. Differential of heat adaptation model, Eq. 13. The res~rltis an initial curvature tending t o horizontal straight line. I
TIME
-
1
FIG. 4. Survivor curve resulting from heat adaptation, showing exponential nature.
parabolic curve derived from the innate heterogeneity theory applied at least for t 5 20/K0, that is, the time required to reduce 20 initial log cycles (20 Do). These two curves have features worth noting. Both have slopes of -KO as t approaches zero. The curve resulting from the heat adaptation theory approaches a straight line at a sufficiently large time. This line has a slope of -Ko(l - ci) and intercepts the ordinate at a value of -aBKo. Thus, if accurate data are available at sufficiently small and large times, a , B a n d K Ocan all be determined graphically.
Quite often, however, data are not adequate to differentiate between the two models by plotting on semi-log paper. Thus, it is difficult t o distinguish visually whether a concave curve is parabolic o r exponential. Whether the experimental survivor curve is parabolic or exponential can be distinguished by comparison of the slope of the survivor curve, dldt log C(t)/Co, as a function of time. Whereas the differential of the parabolic curve (Eq. 5) produces a straight line with slope a', as shown in Fig. 5, the differential of the exponential curve (Eq. 13) produces the curve shown in Fig. 6. This approach has been applied to the data of Figs. 1 and 2 with results appearing in Fig. 7. Data points are indicated; the curves result from use of the appropriate parameter values sub-
Can. J. Microbiol. Downloaded from www.nrcresearchpress.com by UNIV WINDSOR on 11/15/14 For personal use only.
C A N . J. MICROBI(3L. VOL. 22. 1976
dt
-14
0 DATA
OF CURVE
@,
0 DATA
OF CURVE
@ , Fng 2
Fse 2
TIME (MIN)
FIG. 7. (A) Differential of 8 0 ° C and 95 "C data o f Fig. 1. (B) Differential of data from curves 1 and 2 of Fig. 2. Curvature confirms that heat adaptation model applies.
stituted in Eq. 13. Data at 85 "C and 90 "C from Fig. 1 give equally good results, but are not included in Fig. 7 for clarity. Data from curve 3 of Fig. 2 were insufficient t o test. Results in Fig. 7 clearly show that the cause of curvilinearity in the survival curves of B. coagulans and B. cereus is the development of heat resistance by the spores during the heating period. In the natural case, where both heterogeneity and heat adaptation simultaneously cause the curvature, a con~promisedcurve results, showing both influences. Usually, one effect is more pronounced than the other. Thus, with the vegetative cells, whose population is believed t o be more heterogeneous than that of spores, survivor curves resemble parabolic more than exponential curves, whereas with the spores, the opposite results. Because heat adaptation by the spores is more likely to occur at lower than a t higher tempera-
tures within the lethal range, the survivor curves at different temperatures also reveal different shapes, showing those more parabolic at higher temperatures but more exponential at lower ones. The mathematical niodels that we have proposed are mainly phenon~enological, and the validity of these models certainly rests o n the experimental data on the destructive mode of bacterial spores. However, once the validity of these models has been established through further experimental evidence, the models will provide a useful way t o describe the thermal-destruction characteristics of each species of bacterium a n d bacterial spores. ABRAMOWITZ, M., and L. A. S E G U N1965. . Handbook of mathematical functions. Dover Publications Inc., New York. N.Y. G., P. A. THOMPSON, and N. S N E L L1964. . ALDERTON, Heat adaptation and ion exchange in Brrci1lrr.s r,ic8gritr.r.i~rttispores. Science, 143: 141-143. F R A N KH. . A,, and L . L. C A M P B E LJLR. . 1957. The nonlogarilhmic rate of thermal destruction of spores of B. corrgrrlrrr~.s.Appl. Microbiol. 5 : 243-248. H A N .Y. W.. H. A. SCHUYTEN. JR., and C. D. C A L L I H A N . 1971. The combined effect of heat and alkali insterilizing sugarcane bagasse. J. Food Sci. 36: 335-338. H A N .Y. W. 1975. Death rates of bacterial spores: nonlinear survivor curves. Can. J . Microbiol. 21: 1464-1467. JOHNSON. F . , H. E Y R I N Gand , M. J. POLISSAR. 1954. The kinetic basis of molecular biology. John Wiley & Sons, Inc., New York. KOMEMUSH S..I . K. OKUBO, and G . T E R U I1968. . Kinetics on the thermal death of microrganisms. IV. On the change of death rate constant of bacterial spores in the course of heat sterilization. J . Ferment. Technol. 46(3): 249-256. M E Y N E L LG., G., and E. M E Y N E L L1970. . Theory and practice in experimental bacteriology. Cambridge University Press. London. MOATS,W. A. 1971. Kinetics of thermal death of bacteria. J . Bacteriol. 105: 165-171. RAHN,0 . 1943. The problems of the logarithmic order of death in bacteria. Biodynamica, 4: 81. STUMBO. C. R. 1965. Thermobacteriology in food processing. Academic Press, lnc.. New York. WOOD.T. H. 1956. Lethal effects of high and low temperatureson ~ i n i c e l l ~ ~organisms. lar Adv. Biol. Med. Phys. 4: 119-165.
