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Analysis of a novel class of predictive microbial growth models and application to coculture growth
International Journal of Food Microbiology 100 (2005) 107 – 124 www.elsevier.com/locate/ijfoodmicro
Analysis of a novel class of predictive microbial growth models and application to coculture growth F. Poscheta, K.M. Vereeckena,1, A.H. Geeraerda, B.M. NicolaRb, J.F. Van Impea,* a
BioTeC-Bioprocess Technology and Control, Department of Chemical Engineering, Katholieke Universiteit Leuven, W. de Croylaan 46, B-3001 Leuven, Belgium b Laboratory for Postharvest Technology, Department of Agro-Engineering and -Economics, Katholieke Universiteit Leuven, W. de Croylaan 42, B-3001 Leuven, Belgium Received 27 September 2004; accepted 6 October 2004
Abstract In this paper, a novel class of microbial growth models is analysed. In contrast with the currently used logistic type models (e.g., the model of Baranyi and Roberts [Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277–294]), the novel model class, presented in Van Impe et al. (Van Impe, J.F., Poschet, F., Geeraerd, A.H., Vereecken, K.M., 2004. Towards a novel class of predictive microbial growth models. International Journal of Food Microbiology, this issue), explicitly incorporates nutrient exhaustion and/or metabolic waste product effects inducing stationary phase behaviour. As such, these novel model types can be extended in a natural way towards microbial interactions in cocultures and microbial growth in structured foods. Two illustrative case studies of the novel model types are thoroughly analysed and compared to the widely used model of Baranyi and Roberts. In a first case study, the stationary phase is assumed to be solely resulting from toxic product inhibition and is described as a function of the pHevolution. In the second case study, substrate exhaustion is the sole cause of the stationary phase. Finally, a more complex case study of a so-called P-model is presented, dealing with a coculture inhibition of Listeria innocua mediated by lactic acid production of Lactococcus lactis. D 2004 Elsevier B.V. All rights reserved. Keywords: Predictive microbiology; Mechanistic modelling; Product inhibition; Substrate depletion; Coculture growth; Lactic acid
1. Introduction * Corresponding author. Tel.: +32 16 32 14 66; fax: +32 16 32 29 91. E-mail address: [email protected] (J.F. Van Impe). URL: http://www.kuleuven.ac.be/cit/biotec/. 1 Present address: Federal Agency for the Safety of the Food Chain, WTC III, Simon Bolivarlaan 30, B-1000 Brussel, Belgium. 0168-1605/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijfoodmicro.2004.10.008
Single species growth can be described with a sigmoid-like curve, which can be considered as a sequence of three distinct phases. During the first phase, the lag phase, the microorganisms adapt to their new environment, and their total number
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remains (approximately) equal to their initial population level. In a subsequent phase, the exponential growth phase, the microorganisms reproduce exponentially. After attaining their maximum population level, the total number of microorganisms remains constant again during the stationary phase (the subsequent phase of cell number decline is not considered in this research). In this paper, a novel class of microbial growth models is discussed, and two limiting case studies of this novel class of predictive microbial growth models are analysed and compared to a generally applied logistic-type model, the model of Baranyi and Roberts (1994). Part of the results of this paper is also reported in Van Impe et al. (2003) and Poschet et al. (2003a).
2. A critical view to the model of Baranyi and Roberts as a prototype of the logistic-type microbial growth model In order to enable a clear comparison with the novel model types, some model simulations and evaluations of the model of Baranyi and Roberts are presented in this section. 2.1. Model description In this paper, the following global implicit formulation of the model of Baranyi and Roberts (Baranyi and Roberts, 1994) valid under dynamic environmental conditions is used: dN ðt Þ Qðt Þ N ðt Þ ¼ lmax d d 1 dN ðt Þ ð1Þ dt 1 þ Qðt Þ Nmax with N(t=0)=N 0 dQðt Þ ¼ lmax dQðt Þ dt
ð2Þ
with Q(t=0)=Q 0. The first differential equation (Eq. (1)) describes the evolution of the microbial load N(t) with N 0 the initial microbial load [CFU/mL]. The first factor in the righthand side of Eq. (1) induces the exponential growth phase with maximum specific growth rate l max [1/h]. The second factor is the adjustment function describ-
ing the lag phase by means of a so-called physiological state Q(t) [], assumed to be proportional to the concentration of a (hypothetical) critical substance simulating the bottle-neck in the growth process. The third factor is the logistic inhibition function that accounts for the stationary phase by means of the maximum microbial load parameter N max [CFU/mL]. The second differential equation (Eq. (2)) describes the exponential evolution of Q(t), with Q 0 [] the initial physiological state. Remark that the adjustment function can be regarded as mechanistically inspired, whereas the logistic-type inhibition function is purely empirical as it does not include any cause–effect relationship (Lynch and Poole, 1979). For static environmental conditions (e.g., constant temperature, pH,. . .), the following analytical solution can be derived: 1 þ Q0 dexpðlmax t Þ N0 Nmax 1 þ Q0 ð3Þ N ðt Þ ¼ 1 þ Q0 dexpðlmax t Þ Nmax N0 þ N0 1 þ Q0 It can easily be verified (by replacing t in Eq. (3) by 0) that the initial microbial load N(t=0) is equal to N 0, as imposed by the initial condition of Eq. (1). Also, the maximum microbial load N(t=l) can easily be calculated (by replacing t in Eq. (3) by l) and is equal to N max. 2.2. Model simulations The influence of the four model parameters (N 0, Q 0, l max, N max) on the microbial growth curve is illustrated in Fig. 1. Model simulations were performed while varying only one of the four model parameters and keeping the other three constant. This procedure was repeated for all four model parameters. The influence of N 0, l max, and N max is self-evident. Q 0 clearly influences the length of the lag phase: the smaller the Q 0, the longer the lag phase. 2.3. Monte Carlo analysis A Monte Carlo analysis was performed on an experimental growth curve of E. coli K12 at 35 8C to investigate the propagation of uncertainty on experimental data to model parameters. More details on the exemplary experimental data can be found in Bernaerts
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Fig. 1. Influence of the Baranyi model parameters on the microbial growth curve. n 0=lnN 0, n max=lnN max, and q 0=lnQ 0.
et al. (2000). The Monte Carlo analysis, as implemented in this research, comprises the following steps: (i) fit of the experimental data set with the model; (ii) considering the model fit as a perfect data set and assuming an additive error function representing variation in the measurements which is expressed by a normal distribution N (0,0.25) (which is a rather conservative assumption) and corresponds to a 95% confidence interval of 1 log10 unit on the experimental data (Jarvis, 1989); (iii) generating a random data set (on the basis of step (ii)), and (iv) fitting the random data set with the model. Steps (iii) and (iv) are repeated 10,000 times, and the resulting parameter distributions
are presented in Fig. 2. More details on the methodology can be found in, e.g., Poschet et al. (2003b). 2.4. Model evaluation The model of Baranyi and Roberts is widely used for a number of reasons: (i) it is easy to use, (ii) it is applicable under dynamic conditions, (iii) it has good fitting capacities, and (iv) it has a microbiologically inspired lag phase and exponential growth phase. Contrary to the adjustment function, the inhibition function is not mechanistically inspired. Although clearly interpretable a
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Fig. 2. Model parameter distributions for the model of Baranyi and Roberts using an exemplary growth curve.
posteriori, this mathematical abstraction, inherited from the logistic model type, lacks a mechanistic base since it does not encapsulate a reason why the microbial population stops growing. In other words, it does not reflect any cause–effect relationship. Another drawback of the model is its ad hoc approach in complex situations (e.g., growth in structured media, cocultures). The latter is illustrated with a coculture growth experiment of Lactococcus lactis and Listeria innocua at 35 8C originating from Vereecken (2002) and Vereecken et al. (2002). In this coculture experiment, the homofermentative L. lactis produces lactic acid. The toxic activity of lactic acid is associated with (i) the lowering of the medium pH and (ii) the simultaneous formation of undissociated lactic acid, which is able to cross the cell membrane and causes an intracellular pH drop (Russel, 1992; Lambert and Stratford, 1999; Vereecken et al., 2003). The growth of L. innocua is monitored for six different initial concentrations of the antagonist L. lactis (see Fig. 3). The model of Baranyi and
Roberts—with following degrees of freedom: lnN 0, lnN max, lnQ 0, and l max—can describe each experimental growth curve of L. innocua separately but—contrary to, e.g., the P-model described below—fails in describing the global phenomenon of an earlier induced stationary phase with increasing initial concentration of the antagonist L. lactis because of the ad hoc value of N max associated with each single growth curve. The model parameter estimations and the Sum of Squared Errors (SSE) of the model description on the six separate growth curves are listed in Table 1. A separate note on the estimation of l max is to be made here. It can be seen that, although the slope of the six growth curves is more or less equal (see also Fig. 6), the estimation of l max is increasing when the N max observed decreases. A decreased N max self-evidently results in a decreased inhibition function (1N/N max). In this situation, the estimated l max increases to compensate for this smaller inhibition function value in order to obtain similar numerical values of the (complete) right-
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Fig. 3. Description of the model of Baranyi and Roberts on experimental data of L. innocua in different coculture experiments. (o): experimental data point, (–): model description. N 0,Li, the initial inoculum size of L. innocua is equal to 103 in all plots. N 0,Ll, the initial inoculum size of L. lactis is variable: upper left plot: monoculture, upper right plot: N 0,Ll=103, middle left plot: N 0,Ll=104, middle right plot: N 0,Ll=105, lower left plot: N 0,Ll=106, lower right plot: N 0,Ll=107.
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Table 1 Model parameter values and SSE for the model of Baranyi and Roberts, the P-model, and the S-model on the six separate growth curves of L. innocua Data set
Model
ln(N 0)
l max
ln( Q 0)
ln(N max)
1
Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model
7.6001 7.6001 7.6001 8.0088 8.0087 8.0087 8.0425 8.0424 8.0424 7.9004 7.9003 7.9003 7.9567 7.9561 7.9561 8.0840 8.0868 8.0868
1.1738 1.1738 1.1738 1.1781 1.1777 1.1777 1.4228 1.4212 1.4212 1.3543 1.3459 1.3459 1.6732 1.6279 1.6279 2.3229 2.0455 2.0455
0.0440 0.0440 0.0440 0.1598 0.1595 0.1595 0.7068 0.7054 0.7054 0.2227 0.2193 0.2193 1.6472 1.6257 1.6257 1.9992 1.9737 1.9737
21.1363
2
3
4
5
6
hand side of Eq. (1). A similar phenomenon is illustrated in the upper left plot of Fig. 1. Although the numerical value of l max is constant for the different simulations, a different value of N 0 results in a different slope of the exponential growth phase, since the value of the inhibition function is different.
3. A novel class of predictive microbial growth models which incorporate more mechanistic knowledge In order to come up with a mechanistically inspired alternative for the logistic model type, a novel class of predictive microbial growth models is presented in Van Impe et al. (2004). The global structure of the novel class of predictive growth models consists of a general expression for the microbial evolution dN ðt Þ ¼ lmax dlQ ðQÞdls ðS ÞdlP ð PÞdN ðt Þ dt
ð4Þ
together with the appropriate differential equations for the physiological state Q [], the substrate(s) S [M], and the toxic product(s) P [M].
ln(K P )
ln( Y N/S )
21.1363 21.1363 16.0495 16.0492 16.0492 14.7198 14.7185 14.7185 12.8709 12.8639 12.8639 11.3036 11.2677 11.2677 9.7593 9.5507 9.5507
SSE 0.4431 0.4431 0.4431 0.3079 0.3079 0.3079 0.2865 0.2866 0.2866 0.1035 0.1034 0.1034 0.6121 0.6119 0.6119 0.6126 0.6089 0.6089
4. Back to basics: a so-called P-model as a first limiting case study of the novel class of microbial growth models 4.1. Model description In a first case study, the stationary phase is assumed to be solely resulting from toxic product inhibition. Mathematically, this implies that the factor l S (S) in Eq. (4) is equal to 1. In Table 2, some product inhibition functions reported in the literature are listed (Vereecken, 2002). In this paper the product inhibition function is assumed to be linear in the product concentration, as proposed in Ghose and Tyagi (1979). Other assump-
Table 2 Examples of product inhibition functions. P max [M]: concentration of P at which growth ceases; a [] shape parameter; k [1/M]: inhibition constant; K [M]: concentration of P for which l P ( P)=1/2 l P ( P) function
Reference
(1(( P)/( P max))) (1(( P)/( P max))a [1-((( P)/( P max)))a] exp(kd P) (((K)/(K+P)))
Ghose and Tyagi (1979) Levenspiel (1980) Luong (1985) Aiba et al. (1968) Aiba and Shocla (1969)
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tions are that (i) the initial concentration of the toxic product P(t=0) is equal to 0, (ii) there is only one growth inhibiting product, and (iii) there is only growth associated production of the toxic product P. The model consists of the following three differential equations: dN ðt Þ Q ðt Þ P ðt Þ ¼ lmax d d 1 dN ðt Þ ð5Þ dt 1 þ Q ðt Þ KP with N(t =0)= N 0 dQðt Þ ¼ lmax dQðt Þ dt
ð6Þ
with Q(t =0)= Q 0 dPðt Þ Q ðt Þ P ðt Þ ¼ YP=N dlmax d d 1 dN ðt Þ dt 1 þ Qðt Þ KP ð7Þ with P(t =0)= 0. The first equation describes the microbial evolution in time and consists of the adjustment function of the model of Baranyi and Roberts (see Eq. (1)) and an inhibition function which is linear in function of the toxic product concentration P [M]. The larger the concentration of product P(t), the smaller the increase in microorganisms. The second equation is equal to the second equation of the model of Baranyi and Roberts (Eq. (2)) and describes the exponential evolution of the physiological state. The third equation describes the evolution of the toxic product concentration P [M], with Y P/N [M/(CFU/mL)] as the yield for product over microorganisms. Dividing Eq. (5) by Eq. (7) results in dN ðt Þ 1 ¼ dPðt Þ YP=N
ð8Þ
Rearranging and integrating with the appropriate boundaries results in the following relation between N(t) and P(t): N ðt Þ ¼ N0 þ
P ðt Þ YP=N
ð9Þ
The inhibition function chosen implies that the maximum value for P is equal to K P . The asymptotic
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microbial load N as is equal to N 0+(KP / Y P/N ). Substituting all this in Eq. (5) results in: dN ðt Þ Q N ðt Þ N0 ¼ lmax d d 1 dN ðt Þ dt 1þQ Nas N0 ð10Þ By comparing Eq. (10) with Eq. (1), it can be concluded that the P-model and the model of Baranyi and Roberts are analytically only slightly different. Since N 0 is negligible in comparison with both N(t) (for increasing time instants) and N as, both models are equal from a numerical point of view. For this model, also a static version can be derived: þKP 1 þ Q0 expðlmax tÞ N0 YP=N KP N0 ðN0 YP=N þ KP Þ 1 þ Q0 N ðt Þ¼ þKP N0 YP=N KP 1 þ Q0 expðlmax tÞ KP þ N0 YP=N 1 þ Q0 ð11Þ The factor N 0Y P/N is negligible in comparison with K P for small values of Y P/N , and by consequence, also, this static model version is analytically only slightly different from the static model version of the model of Baranyi and Roberts (see Eq. (3)). Remark that the initial microbial load and the asymptotic microbial load can also be calculated by replacing t in Eq. (11) by 0 or by l, respectively. 4.2. Model simulation A simulation study reveals the influence of the different model parameters (N 0, Q 0, l max, K P, and Y P/N ) on the microbial growth curve. The results are depicted in Fig. 4. In the upper left plot of Fig. 4, it is clearly shown that a high initial microbial load N 0 also increases the asymptotic microbial load. Although this influence is not well known (and is not present in the model of Baranyi and Roberts), it is in compliance with experimental data discussed in Carlin et al. (1995). Carlin et al. (1995) investigated the factors affecting the growth of L. innocua and reported that the population reached after 7 days (which can be considered the maximum population level) was lower for low inocula (10–103 CFU/g) compared to high inocula (105 CFU/g). This effect can be deducted by means of Eq. (9): N 0 influences the asymptotic value
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Fig. 4. Influence of the different P-model parameters on the microbial growth curve. n 0=lnN 0, k=lnK P, and q 0=lnQ 0.
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of the microbial load concentration. The upper middle and right plots are similar to the corresponding ones of the model of Baranyi and Roberts. In the lower left plot, it can be seen that K P corresponds to the asymptotic microbial load for high values of K P but not for low values. This is because for small values of K P, the influence of N 0 is not negligible on the asymptotic microbial load (see also Eq. (9)). In the lower right plot, the influence of Y P /N on the asymptotic microbial load is presented. For increasing values of Y P /N , the asymptotic microbial load decreases according to Eq. (9). 4.3. Monte Carlo analysis The results of the Monte Carlo analysis (performed as described in Section 2.3) are depicted in Fig. 5. In order to have the same number of model parameters as the model of Baranyi and Roberts, the yield Y P/N is assumed to be equal to 1 (it can easily be verified that this is always possible). When comparing Figs. 2 and 5), it can be concluded that the model parameter
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distributions, induced by experimental variation on the data, are similar for both the model of Baranyi and Roberts and this prototype P-model. 4.4. Model evaluation 4.4.1. From a P-model numerically equivalent to the Baranyi and Roberts model. . . In order to illustrate the numerical equivalence of the P-model with the model of Baranyi and Roberts, the previously mentioned L. innocua–L. lactis coculture was also described using Eq. (11) with following degrees of freedom: lnN 0, lnK P, lnQ 0, and l max. There is no visual difference between the model description of the model of Baranyi and Roberts (Fig. 3) and the P-model, and therefore, the latter is not presented in a figure; the resulting SSE and model parameter estimations are also listed in Table 1. The P-model results in equal SSE and parameter values (with N max numerically equal to N 0+K P /Y N/P ). The model of Baranyi and Roberts needs six different values of N max to describe the six different growth curves (see
Fig. 5. Model parameter distributions for the P-model using an exemplary growth curve.
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Fig. 3). This version of the P-model, described here solely for the sake of illustrating numerical equivalence with the model of Baranyi and Roberts (i.e., equal fitting properties), obviously needs six different values of K P to describe the six growth curves. 4.4.2. . . . to a more mechanistic P-model In order to encapsulate the biological mechanism for transition from the exponential phase to the stationary phase, i.e., the acidification of the medium because of the production of lactic acid, a more elaborated P-model was developed. The proton formation, which is quantitatively different for each of the six distinct growth curves, is considered as toxic product formation in this case study. The pHmeasurements (taken from Vereecken, 2002 and Vereecken et al., 2002, not shown) are used as an input for the model (i.e., the measurements are directly included in the third factor of the right-hand side of Eq. (12)). The use of the pH-measurements is equivalent to the use of the lactic acid concentrations, since for a given medium, there is a 1–1 relation between the lactic acid concentration and the pH. Mathematically, the model transformation is expressed by the following model reparameterisation: P(t)=[H +](t)=10pH(t). K P is also reparameterised to pHmin which is equal to log10(K P ). The model equations are then as follows: dN ðt Þ Qðt Þ 10pH ðtÞ ¼ lmax d d 1 pH dN ðt Þ dt 1 þ Qðt Þ 10 min ð12Þ with N(t=0)=N 0 dQðt Þ ¼ lmax d Qðt Þ dt
ð13Þ
with Q(t=0)=Q 0. Since this model incorporates the measurements of the pH-evolution (which is different for each of the six distinct growth curves and which is directly included in Eq. (12)), this model only needs a single value of pHmin (and thus of K P ) to describe all six growth curves adequately at once, whereas the model of Baranyi and Roberts needs six different values of N max to describe all six growth curves. The underlying cause of the (early induced) stationary phase, the production of lactic acid, here, modelled by the
Fig. 6. Description of the P-model on experimental data of L. innocua in different coculture experiments. The initial inoculum size of L. innocua is equal to 103 for all experiments. N 0,Ll, the initial inoculum size of L. lactis is variable: (o): monoculture, (+): N 0,Ll=103, (): N 0,Ll=104, (*): N 0,Ll=105, (D): N 0,Ll=106, (q): N 0,Ll=107.
evolution of the pH, quantitatively differs for each of the six growth curves. However, its influence on the evolution of L. innocua can be described by means of one global parameter pHmin . As a result, the total set of six growth curves is described by means of four global model parameters: ln(N 0)=7.9463 ln(CFU/mL), l max=1.5492 h1, ln( Q 0)=0.4553, and pHmin=5.6374. The biological interpretation of pHmin is clear: it is the minimum pH of growth of L. innocua. The model description on all six growth curves at once is presented in Fig. 6.
5. Back to basics: a so-called S -model as a second limiting case study of the novel class of microbial growth models 5.1. Model description In a second special case of the general model (Eq. (4)), the stationary phase is assumed to be only a result of substrate exhaustion and not of toxic product inhibition. Mathematically, this implies that the factor l P ( P) is assumed to be equal to 1. For this case study, it is also assumed that (i) a linear relation is appropriate to describe the influence of substrate depletion on the microbial growth, (ii) there is no substrate consumption for maintenance, (iii) there is
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no substrate breakdown in the medium, (iv) no additional substrate is added during the growth process, and (v) there is only one limiting substrate. These assumptions result in the following three differential equations:
Rearranging and integrating with the appropriate boundaries results in the following relation between N(t) and S(t):
dN ðt Þ Q ðt Þ ¼ lmax d dS ðt ÞdN ðt Þ dt 1 þ Q ðt Þ
The asymptotic microbial load N as is then equal to N 0+Y N/S . Substituting all this in Eq. (14) results in:
ð14Þ
N ðt Þ ¼ N0 þ YN =S d ð1 S ðt ÞÞ
ð18Þ
dN ðt Þ Q N ðt Þ N0 ¼ lmax d d 1 dN ðt Þ dt 1þQ Nas N0
with N(t =0)= N 0
ð19Þ dQðt Þ ¼ lmax d Qðt Þ dt
ð15Þ
with Q(t=0)= Q 0 dS ðt Þ Qðt Þ S ðt Þ ¼ lmax d dN ðt Þ d dt 1 þ Qðt Þ YN =S
ð16Þ
with S(t =0) =S 0. The first equation (Eq. (14)) describes the microbial evolution in time and consists of the adjustment function of the model of Baranyi and Roberts (see Eq. (1)) and an inhibition function which is, as an example, selected equal to the (rescaled) substrate concentration S. Eq. (15) is equal to the second equation of the model of Baranyi and Roberts (Eq. (2)) and describes the exponential evolution of the physiological state. The third equation (Eq. (16)) describes the evolution (consumption) of the substrate concentration S. By consequence, the right-hand side of Eq. (16) is, except for the minus sign and the yield coefficient Y N/S , equal to the right-hand side of Eq. (14). The substrate concentration is rescaled in order to obtain a strictly monotone decreasing inhibition function with values between 1 and 0 and to have a dimensionless substrate concentration. Alternative substrate inhibition functions, inspired on, e.g., Monod kinetics, would obviously also be possible (with other assumptions). Dividing the first model equation (Eq. (14)) by the third one (Eq. (16)) results in dN ðt Þ ¼ YN =S dS ðt Þ
ð17Þ
Eq. (19) is equal to Eq. (10), and after comparison with Eq. (1), it can be concluded that also the S-model and the model of Baranyi and Roberts are analytically only slightly different. Since N 0 is negligible in comparison with both N(t) (for increasing time instants) and N as, both models are equal from a numerical point of view. For this model, a static version can be derived:
N0 þYN =S 1 þ Q0 expðlmax tÞ YN =S N0 ðN0 þ YN =S Þ 1 þ Q0 N ðt Þ ¼ N0 þYN=S 1 þ Q0 expðlmax tÞ YN =S YN =S þ N0 1 þ Q0 ð20Þ Again, this static model version analytically slightly differs from the static model version of the model of Baranyi and Roberts. Since N 0 is negligible compared to Y N/S , both models can be considered as numerically equal. The initial microbial load N(t=0), which is calculated by replacing t in Eq. (20) by 0, is equal to N 0. 5.2. Model simulation A simulation study reveals the influence of the different model parameters (N 0, Q 0, l max, Y N/S , and S 0) on the microbial growth curve. The results are depicted in Fig. 7. In the upper left plot of Fig. 7, it is clearly shown that a high initial microbial load N 0 increases the asymptotic microbial load. Again, this is in compliance with experimental the data presented in Carlin et al. (1995) and can be deducted by means of Eq. (18): N 0 influences the
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Fig. 7. Influence of the different S-model parameters on the microbial growth curve. n 0=lnN 0, y=lnY N/S , and q 0=lnQ 0.
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(asymptotic) value of the microbial load concentration. The upper middle and right plots are similar to the corresponding ones of the model of Baranyi and Roberts. In the lower left plot, it can be seen that Y N/S corresponds to the asymptotic microbial load for high values of Y N/S but not for low values. This is also explained by interpretation of Eq. (18): for a high value of Y N/S , the contribution of N 0 to the (asymptotic) value of the microbial load concentration is negligible. The lower right plot illustrates the influence of S 0 on the asymptotic microbial load, also experimentally reported in Leroy et al. (2003) who investigated the influence of the initial glucose concentration, amongst others, on the maximum biomass concentration of Enterococcus faecium RZS C5. S 0 also influences the maximum specific growth rate. Both influences are also reported for exciting bacteria in Verluyten et al. (2002). The influence of the initial substrate concentration S 0 on the maximum microbial load population is also depicted in the left part of Fig. 5, Van Impe et al. (2004).
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5.3. Monte Carlo analysis The results of the Monte Carlo analysis (performed as described in Section 2.3) are depicted in Fig. 8. In order to have four model parameters as in the model of Baranyi and Roberts, it is assumed that S(t =0) = S 0 is equal to 1. A comparison of Figs. 2 and 8 reveals that the model parameter distributions are similar for the model of Baranyi and Roberts and the S-model. 5.4. Model evaluation Although substrate S is not the limiting factor in this case, the separate growth curves of the previously mentioned L. innocua–L. lactis coculture can also be described with Eq. (20) to illustrate the equal fitting capacity of the S-model vs. the model of Baranyi and Roberts. There is no visual difference between the model description of the model of Baranyi and Roberts (Fig. 3) and the S-model, and therefore, the latter is not presented in a figure; SSE and model
Fig. 8. Model parameter distributions for the S-model using an exemplary growth curve.
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parameter estimations are also listed in Table 1. The S-model results in equal SSE and parameter values (with N max numerically equal to N 0+Y N/S ). Selfevidently, it is not claimed that this ad hoc procedure would be more suitable than the model of Baranyi and Roberts, and for case studies where substrate is a limiting factor, (advanced) substrate measurements are mandatory for a sound application of these S-type models. An example of such an S-type model is presented by Leroy and De Vuyst (2001). In that research, the growth inhibition was represented as a combined effect of lactic acid inhibition, sugar limitation, and nutrient depletion.
concentration for organism i. is selected equal to 10–6. This small value is needed from a numerical point of view to facilitate the estimation of parameters [LaH]max,i,G and [H +]max,i,G . The above equation, introduced in Vereecken et al. (2003), describes the growth suppression by both pH and undissociated lactic acid on both organisms, with the target being the more sensitive one. A second (biological) part of the model describes the production of lactic acid by both organisms:
6. A more elaborated P-model for microbial interaction
mLaH;i ðLaH Þ ¼ 1
In this final section, a more realistic (and more complex) P-model describing the interaction between a target organism, L. innocua, and a microbial antagonist, the lactic acid bacterium L. lactis, is presented. The inhibition of the target organism is explicitly related to the lactic acid production curve. Both inhibitory effects of lactic acid—the lowering of the medium pH and the formation of undissociated lactic acid (as mentioned above)—are included in the model. This model can by consequence be seen as a further extension of the model presented in Section 4.4.2. The model originates from Vereecken (2002). A first (biological) part of the model considers the growth of target (i=1) and antagonist (i=2): dNi ðt Þ Qi ðt Þ ¼ li dNi ðt Þ ¼ lmax;id dlLaH;i dNi ðt Þ dt 1 þ Qi ðt Þ
lLaH;i
¼ 1
½LaH ½LaH max;i;G
1þ d 1
½H þ ½H þ max;i;G
1þ
ð21Þ with N i (t=0)=N 0,i with [H +] [M] the proton concentration, [LaH] [M] the undissociated lactic acid concentration, [LaH]max,i,G [M] and [H +]max,i,G [M] the growth inhibiting lactic acid concentration and the growth inhibiting proton
2 X dLaH tot ¼ ðYLaH=Ni dli þ mmax;i dmLaH;i ðLaH ÞÞdNi dt i¼1
½LaH ½LaH max;i;M
1
1þ
½H þ ½H þ max;i;M
1þ ð22Þ
with LaHtot [M] the total lactic acid concentration, Y LaH/N i [mmol/CFU] the yield of lactic acid for organism i, m max,i [mmol/(CFU h)] the maintenance coefficient of organism i, and [LaH]max,i,M [M] and [H +]max,i,M [M] the metabolism inhibiting lactic acid concentration and the metabolism inhibiting proton concentration for organism i. This equation is proposed in Vereecken et al. (2002). The last (chemical) part concerns the dissociation kinetics of lactic acid in the medium, for which the buffering capacities are not exactly known. [LaH] and pH are computed in accordance with the method described in Vereecken and Van Impe (2002). In order to represent the relation between [LaH] and LaHtot , the following relation, inspired by Dabeskinetics, is proposed. ab ½LaH ¼ aLaH tot ðLaH tot þ bÞ 2ð b c Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðLaH tot þ bÞ2 4ðb cÞLaH tot ð23Þ An interpretation of the model parameters a, b, and c is presented in Vereecken and Van Impe (2002). To
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Fig. 9. Experimental data and model description (Eqs. (20)–(23)) of the L. lactis–L. innocua coculture. (o): cell concentration of the antagonist, (w): cell concentration of the target (: data point of the target below detection limit), (D): lactic acid concentration, (*): pH. Dash-dotted line: simulation beyond experimentally observed stationary phase.
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represent the relation between pH and [LaH], a second expression, also based on the Dabes-kinetics, is developed as follows. pH ¼
1 ½ðb 2c1 Þ½LaH 2a1 c1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 ½LaH2 þ 4a1 b21 c1 ½LaH þ pH 0
ð24Þ
Model parameters a 1, b 1, and c 1 have a similar interpretation as a, b, and c, respectively, and which can also be found in Vereecken and Van Impe (2002). pH0 symbolises the initial pH. Eqs. (23) and (24) determine the medium-related parameters. The procedure should be accomplished once for a specific medium and initial pH. Indeed, since the model parameters a, b, c, a 1, b 1, and c 1 only depend on the medium characteristics and pH0 only on the initial pH, they are suited for every growth experiment in this particular medium and initial pH. After calibration, prediction of the time dependent course of pH (and [LaH]) is always possible when a only LaHtot profile is measured. Application of the complete model to the experimental data set of cell concentrations N i , total lactic acid concentration LaHtot, and pH consists out of two major steps: (i) determination of the medium related parameters (i.e., the parameters of Eqs. (23) and (24)) and (ii) determination of growth and production related parameters (i.e., the parameters of Eqs. (21) and (22)). In the first step, the [LaHtot(t), pH(t)] measurements are the input to estimate the model parameters of Eqs. (23) and (24), without considering the cell concentration data. To identify the parameters of Eqs. (21) and (22), describing the cell growth and metabolite production, the complete data set (including cell counts, lactic acid, and pH measurements) is necessary. Eqs. (23) and (24) are used to determine the input values [LaH](t) and [H +](t) of the specific growth and production rates (Eqs. (21) and (22)). Fig. 9 represents the experimental data points of cell concentration, total lactic acid production and pH, and the highly accurate description with the model (Eqs. (21)–(24)) for three experiments of the L. innocua–L. lactis coculture at 35 8C, namely, the monocultures and one coculture (with a L. lactis inoculum level of 105 CFU/ mL). The model contains 14 model parameters: four
parameters of growth, three parameters for lactic acid production, and seven parameters for the lactic acid dissociation. More details can be found in Vereecken (2002). For each individual growth (monoculture/ coculture) curve, the onset of the stationary phase coincides with the large rate of increase of lactic acid production. In the coculture experiments, two antagonistic effects emerge, namely, the early induction of the stationary of the target and a decline phase, where the target’s cell concentration is reduced to below detection limit. Note that, in view of the model equations, only the inhibiting effect of lactic acid is considered (i.e., the induction of the stationary phase) and not the inactivation effect (explaining the decline phase).
7. Conclusions The main contribution of this paper is the analysis of a novel class of predictive microbial growth models which reflect the (micro)biological phenomena governing the microbial growth process. This research particularly focuses on the transition from the exponential growth phase to the stationary phase, which is generally assumed to be the result of substrate depletion and/or toxic product inhibition. Two limiting case studies of the novel class of microbial growth models are carefully analysed and compared to the model of Baranyi and Roberts, the currently most used logistic-type model. Both the socalled P-model and S-model (i) have an equal fitting capacity as the model of Baranyi and Roberts, (ii) have identical model parameter distributions, (iii) are easier to extend to more complex situations, and (iv) are applicable to both the macroscopic (i.e., population) and the microscopic (i.e., individual cell) levels. A more complex P-model describing the interaction between a target organism and an antagonist through the formation of lactic acid by the antagonist is presented to illustrate the power of the novel class of models.
8. Uncited references Bailey and Ollis, 1986 Ciftci et al., 1983
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Foegeding, 1997 ICMSF, 1988 McMeekin et al., 1992 McMeekin et al., 1997 Motarjemi and KRaferstein, 1999 Ross, 1999 Zwietering and van Gerwen, 2000
Acknowledgements This research is supported by the Research Council of the Katholieke Universiteit Leuven as part of project IDO/00/008, the Institute for the Promotion of Innovation by Science and Technology (IWT), the Fund for Scientific Research– Flanders (FWO) as part of project G.0213.02 and for the Postdoctoral Fellowship of AG, the Belgian Program on Interuniversity Poles of Attraction and the Second Multiannual Scientific Support Plan for a Sustainable Development Policy, initiated by the Belgian Federal Science Policy Office, and the European Commission as part of project QLK1-CT2001-01415. The scientific responsibility is assumed by its authors.
References Aiba, S., Shocla, M., 1969. Reassessment of product inhibition in alcohol fermentations. Journal of Fermentation Technology 47, 790 – 798. Aiba, S., Shocla, M., Nagatani, M., 1968. Kinetics of product inhibition in alcohol fermentation. Biotechnology and Bioengineering 10, 845 – 864. Bailey, J.E., Ollis, D.F., 1986. Biochemical Engineering Fundamentals, Second edition. McGraw-Hill, New York. Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277 – 294. Bernaerts, K., Versyck, K.J., Van Impe, J.F., 2000. On the design of optimal dynamic experiments for parameter estimation of a Ratkowsky-type growth kinetics at suboptimal temperatures. International Journal of Food Microbiology 54, 27 – 38. Carlin, F., Nguyen-the, C., Abreu da Silva, A., 1995. Factors affecting the growth of Listeria monocytogenes on minimally processed fresh endive. Journal of Applied Bacteriology 78, 636 – 646. Ciftci, T., Constantinides, A., Wang, S.S., 1983. Optimization of conditions and cell feeding procedure for alcohol fermentation. Biotechnology and Bioengineering 25, 2007 – 2023.
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Foegeding, P.M., 1997. Driving predictive modelling on a risk assessment path for enhanced food safety. International Journal of Food Microbiology 36, 87 – 95. Ghose, T.K., Tyagi, R.D., 1979. Rapid ethanol fermentation of cellulose hydrolysate: II. Product and substrate inhibition and optimization of fermentor design. Biotechnology and Bioengineering 21, 1401 – 1420. ICMSF, 1988. Micro-organisms in foods 4. Application of the Hazard Analysis Critical Control Point (HACCP) System to Ensure Microbiological Safety and Quality. Blackie Academic and Professional, London. Jarvis, B., 1989. Statistical aspects of the microbiological analysis of foods. Progress in Industrial Microbiology, vol. 21. Elsevier, Amsterdam. Lambert, R.J., Stratford, M., 1999. Weak-acid preservatives: modelling microbial inhibition and response. Journal of Applied Microbiology 86, 157 – 164. Leroy, F., De Vuyst, L., 2001. Growth of the bacteriocin-producing Lactobacillus sakei strain CTC 494 in MRS broth is strongly reduced due to nutrient exhaustion: a nutrient depletion model for the growth of lactic acid bacteria. Applied and Environmental Microbiology 6 (10), 4407 – 4413. Leroy, F., Vankrunkelsven, S., De Greef, J., De Vuyst, L., 2003. The stimulating effect of a harsh environment on the bacteriocin activity by Enterococcus faecium RZS C5 and dependency on the environmental stress factor used. International Journal of Food Microbiology 83, 27 – 38. Levenspiel, O., 1980. The Monod equation: a revisit and a generalization to product inhibition situations. Biotechnology and Bioengineering 22, 1671 – 1687. Luong, J.H., 1985. Kinetics of ethanol inhibition in alcohol fermentation. Biotechnology and Bioengineering 27, 280 – 285. Lynch, J.M., Poole, N.J, 1979. Microbial Ecology. Blackwell, London. McMeekin, T.A., Ross, T., Olley, J., 1992. Application of predictive microbiology to assure the quality and safety of fish and fish products. International Journal of Food Microbiology 15, 13 – 32. McMeekin, T.A., Brown, J., Krist, K., Miles, D., Neumeyer, K., Nichols, D.S., Olley, J., Presser, K., Ratkowsky, D.A., Ross, T., Salter, M., Soostranon, S., 1997. Quantitative microbiology: a basis for food safety. Emerging Infectious Diseases 3 (4), 541 – 549. Motarjemi, Y., K7ferstein, F., 1999. Food safety, hazard analysis and critical control points and the increase in foodborne diseases: a paradox? Food Control 10, 325 – 333. Poschet, F., Geeraerd, A.H., Vereecken, K.M., NicolaR, B.M., Van Impe, J.F., 2003a. Analysis of a novel predictive microbial growth model. In: Van Impe, J.F.M., Geeraerd, A.H., Legue´rinel, I., Mafart, P. (Eds.), Predictive Modelling in Foods-Conference Proceedings. KULeuven/BioTeC, Belgium, ISBN: 90-5682-4007. pp. 93 – 95. (Fourth International Conference on Predictive Modelling in Foods, Quimper (France), June 15–19, 2003). Poschet, F., Bernaerts, K., Geeraerd, A.H., Scheerlinck, N., NicolaR, B.M., Van Impe, J.F., 2003b. Monte Carlo analysis as a tool to incorporate variation on experimental data in predictive microbiology. Food Microbiology 20 (3), 285 – 295.
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Ross, T., 1999. Predictive Microbiology for the Meat Industry. Meat and Livestock Australia, North Sydney. Russel, J.B., 1992. Another explanation of the toxicity of fermentation acids at low pH: anion accumulation versus uncoupling. Journal of Applied Bacteriology 73, 363 – 370. Van Impe, J.F., Poschet, F., Vereecken, K.M., Geeraerd, A.H., 2003. Towards a novel class of predictive microbial growth models. In: Van Impe, J.F.M., Geeraerd, A.H., Legue´rinel, I., Mafart, P. (Eds.), Predictive Modelling in Foods—Conference Proceedings. KULeuven/BioTeC, Belgium, ISBN: 90-5682-400-7. pp. 73 – 74 (keynote paper) Fourth International Conference on Predictive Modelling in Foods, Quimper (France), June 15–19, 2003). Van Impe, J.F., Poschet, F., Geeraerd, A.H., Vereecken, K.M., 2004. Towards a novel class of predictive microbial growth models. International Journal of Food Microbiology 100, 97–105 (this issue). Vereecken, K., 2002. Predictive microbiology for multiple species systems: design and validation of a model for lactic acid induced interaction in complex media. PhD Thesis. Faculty of Applied Sciences, Katholieke Universiteit Leuven. Belgium. Vereecken, K., Van Impe, J.F., 2002. Analysis and practical implementation of a model for combined growth and metabolite production of lactic acid bacteria. International Journal of Food Microbiology 73, 239 – 250. Vereecken, K.M., Antwi, M., Janssen, M., Holvoet, A., Devlieghere, F., Debevere, J., Van Impe, J.F., 2002. Biocontrol of
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Analysis of a novel class of predictive microbial growth models and application to coculture growth F. Poscheta, K.M. Vereeckena,1, A.H. Geeraerda, B.M. NicolaRb, J.F. Van Impea,* a
BioTeC-Bioprocess Technology and Control, Department of Chemical Engineering, Katholieke Universiteit Leuven, W. de Croylaan 46, B-3001 Leuven, Belgium b Laboratory for Postharvest Technology, Department of Agro-Engineering and -Economics, Katholieke Universiteit Leuven, W. de Croylaan 42, B-3001 Leuven, Belgium Received 27 September 2004; accepted 6 October 2004
Abstract In this paper, a novel class of microbial growth models is analysed. In contrast with the currently used logistic type models (e.g., the model of Baranyi and Roberts [Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277–294]), the novel model class, presented in Van Impe et al. (Van Impe, J.F., Poschet, F., Geeraerd, A.H., Vereecken, K.M., 2004. Towards a novel class of predictive microbial growth models. International Journal of Food Microbiology, this issue), explicitly incorporates nutrient exhaustion and/or metabolic waste product effects inducing stationary phase behaviour. As such, these novel model types can be extended in a natural way towards microbial interactions in cocultures and microbial growth in structured foods. Two illustrative case studies of the novel model types are thoroughly analysed and compared to the widely used model of Baranyi and Roberts. In a first case study, the stationary phase is assumed to be solely resulting from toxic product inhibition and is described as a function of the pHevolution. In the second case study, substrate exhaustion is the sole cause of the stationary phase. Finally, a more complex case study of a so-called P-model is presented, dealing with a coculture inhibition of Listeria innocua mediated by lactic acid production of Lactococcus lactis. D 2004 Elsevier B.V. All rights reserved. Keywords: Predictive microbiology; Mechanistic modelling; Product inhibition; Substrate depletion; Coculture growth; Lactic acid
1. Introduction * Corresponding author. Tel.: +32 16 32 14 66; fax: +32 16 32 29 91. E-mail address: [email protected] (J.F. Van Impe). URL: http://www.kuleuven.ac.be/cit/biotec/. 1 Present address: Federal Agency for the Safety of the Food Chain, WTC III, Simon Bolivarlaan 30, B-1000 Brussel, Belgium. 0168-1605/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ijfoodmicro.2004.10.008
Single species growth can be described with a sigmoid-like curve, which can be considered as a sequence of three distinct phases. During the first phase, the lag phase, the microorganisms adapt to their new environment, and their total number
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remains (approximately) equal to their initial population level. In a subsequent phase, the exponential growth phase, the microorganisms reproduce exponentially. After attaining their maximum population level, the total number of microorganisms remains constant again during the stationary phase (the subsequent phase of cell number decline is not considered in this research). In this paper, a novel class of microbial growth models is discussed, and two limiting case studies of this novel class of predictive microbial growth models are analysed and compared to a generally applied logistic-type model, the model of Baranyi and Roberts (1994). Part of the results of this paper is also reported in Van Impe et al. (2003) and Poschet et al. (2003a).
2. A critical view to the model of Baranyi and Roberts as a prototype of the logistic-type microbial growth model In order to enable a clear comparison with the novel model types, some model simulations and evaluations of the model of Baranyi and Roberts are presented in this section. 2.1. Model description In this paper, the following global implicit formulation of the model of Baranyi and Roberts (Baranyi and Roberts, 1994) valid under dynamic environmental conditions is used: dN ðt Þ Qðt Þ N ðt Þ ¼ lmax d d 1 dN ðt Þ ð1Þ dt 1 þ Qðt Þ Nmax with N(t=0)=N 0 dQðt Þ ¼ lmax dQðt Þ dt
ð2Þ
with Q(t=0)=Q 0. The first differential equation (Eq. (1)) describes the evolution of the microbial load N(t) with N 0 the initial microbial load [CFU/mL]. The first factor in the righthand side of Eq. (1) induces the exponential growth phase with maximum specific growth rate l max [1/h]. The second factor is the adjustment function describ-
ing the lag phase by means of a so-called physiological state Q(t) [], assumed to be proportional to the concentration of a (hypothetical) critical substance simulating the bottle-neck in the growth process. The third factor is the logistic inhibition function that accounts for the stationary phase by means of the maximum microbial load parameter N max [CFU/mL]. The second differential equation (Eq. (2)) describes the exponential evolution of Q(t), with Q 0 [] the initial physiological state. Remark that the adjustment function can be regarded as mechanistically inspired, whereas the logistic-type inhibition function is purely empirical as it does not include any cause–effect relationship (Lynch and Poole, 1979). For static environmental conditions (e.g., constant temperature, pH,. . .), the following analytical solution can be derived: 1 þ Q0 dexpðlmax t Þ N0 Nmax 1 þ Q0 ð3Þ N ðt Þ ¼ 1 þ Q0 dexpðlmax t Þ Nmax N0 þ N0 1 þ Q0 It can easily be verified (by replacing t in Eq. (3) by 0) that the initial microbial load N(t=0) is equal to N 0, as imposed by the initial condition of Eq. (1). Also, the maximum microbial load N(t=l) can easily be calculated (by replacing t in Eq. (3) by l) and is equal to N max. 2.2. Model simulations The influence of the four model parameters (N 0, Q 0, l max, N max) on the microbial growth curve is illustrated in Fig. 1. Model simulations were performed while varying only one of the four model parameters and keeping the other three constant. This procedure was repeated for all four model parameters. The influence of N 0, l max, and N max is self-evident. Q 0 clearly influences the length of the lag phase: the smaller the Q 0, the longer the lag phase. 2.3. Monte Carlo analysis A Monte Carlo analysis was performed on an experimental growth curve of E. coli K12 at 35 8C to investigate the propagation of uncertainty on experimental data to model parameters. More details on the exemplary experimental data can be found in Bernaerts
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Fig. 1. Influence of the Baranyi model parameters on the microbial growth curve. n 0=lnN 0, n max=lnN max, and q 0=lnQ 0.
et al. (2000). The Monte Carlo analysis, as implemented in this research, comprises the following steps: (i) fit of the experimental data set with the model; (ii) considering the model fit as a perfect data set and assuming an additive error function representing variation in the measurements which is expressed by a normal distribution N (0,0.25) (which is a rather conservative assumption) and corresponds to a 95% confidence interval of 1 log10 unit on the experimental data (Jarvis, 1989); (iii) generating a random data set (on the basis of step (ii)), and (iv) fitting the random data set with the model. Steps (iii) and (iv) are repeated 10,000 times, and the resulting parameter distributions
are presented in Fig. 2. More details on the methodology can be found in, e.g., Poschet et al. (2003b). 2.4. Model evaluation The model of Baranyi and Roberts is widely used for a number of reasons: (i) it is easy to use, (ii) it is applicable under dynamic conditions, (iii) it has good fitting capacities, and (iv) it has a microbiologically inspired lag phase and exponential growth phase. Contrary to the adjustment function, the inhibition function is not mechanistically inspired. Although clearly interpretable a
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Fig. 2. Model parameter distributions for the model of Baranyi and Roberts using an exemplary growth curve.
posteriori, this mathematical abstraction, inherited from the logistic model type, lacks a mechanistic base since it does not encapsulate a reason why the microbial population stops growing. In other words, it does not reflect any cause–effect relationship. Another drawback of the model is its ad hoc approach in complex situations (e.g., growth in structured media, cocultures). The latter is illustrated with a coculture growth experiment of Lactococcus lactis and Listeria innocua at 35 8C originating from Vereecken (2002) and Vereecken et al. (2002). In this coculture experiment, the homofermentative L. lactis produces lactic acid. The toxic activity of lactic acid is associated with (i) the lowering of the medium pH and (ii) the simultaneous formation of undissociated lactic acid, which is able to cross the cell membrane and causes an intracellular pH drop (Russel, 1992; Lambert and Stratford, 1999; Vereecken et al., 2003). The growth of L. innocua is monitored for six different initial concentrations of the antagonist L. lactis (see Fig. 3). The model of Baranyi and
Roberts—with following degrees of freedom: lnN 0, lnN max, lnQ 0, and l max—can describe each experimental growth curve of L. innocua separately but—contrary to, e.g., the P-model described below—fails in describing the global phenomenon of an earlier induced stationary phase with increasing initial concentration of the antagonist L. lactis because of the ad hoc value of N max associated with each single growth curve. The model parameter estimations and the Sum of Squared Errors (SSE) of the model description on the six separate growth curves are listed in Table 1. A separate note on the estimation of l max is to be made here. It can be seen that, although the slope of the six growth curves is more or less equal (see also Fig. 6), the estimation of l max is increasing when the N max observed decreases. A decreased N max self-evidently results in a decreased inhibition function (1N/N max). In this situation, the estimated l max increases to compensate for this smaller inhibition function value in order to obtain similar numerical values of the (complete) right-
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Fig. 3. Description of the model of Baranyi and Roberts on experimental data of L. innocua in different coculture experiments. (o): experimental data point, (–): model description. N 0,Li, the initial inoculum size of L. innocua is equal to 103 in all plots. N 0,Ll, the initial inoculum size of L. lactis is variable: upper left plot: monoculture, upper right plot: N 0,Ll=103, middle left plot: N 0,Ll=104, middle right plot: N 0,Ll=105, lower left plot: N 0,Ll=106, lower right plot: N 0,Ll=107.
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Table 1 Model parameter values and SSE for the model of Baranyi and Roberts, the P-model, and the S-model on the six separate growth curves of L. innocua Data set
Model
ln(N 0)
l max
ln( Q 0)
ln(N max)
1
Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model Baranyi P-model S-model
7.6001 7.6001 7.6001 8.0088 8.0087 8.0087 8.0425 8.0424 8.0424 7.9004 7.9003 7.9003 7.9567 7.9561 7.9561 8.0840 8.0868 8.0868
1.1738 1.1738 1.1738 1.1781 1.1777 1.1777 1.4228 1.4212 1.4212 1.3543 1.3459 1.3459 1.6732 1.6279 1.6279 2.3229 2.0455 2.0455
0.0440 0.0440 0.0440 0.1598 0.1595 0.1595 0.7068 0.7054 0.7054 0.2227 0.2193 0.2193 1.6472 1.6257 1.6257 1.9992 1.9737 1.9737
21.1363
2
3
4
5
6
hand side of Eq. (1). A similar phenomenon is illustrated in the upper left plot of Fig. 1. Although the numerical value of l max is constant for the different simulations, a different value of N 0 results in a different slope of the exponential growth phase, since the value of the inhibition function is different.
3. A novel class of predictive microbial growth models which incorporate more mechanistic knowledge In order to come up with a mechanistically inspired alternative for the logistic model type, a novel class of predictive microbial growth models is presented in Van Impe et al. (2004). The global structure of the novel class of predictive growth models consists of a general expression for the microbial evolution dN ðt Þ ¼ lmax dlQ ðQÞdls ðS ÞdlP ð PÞdN ðt Þ dt
ð4Þ
together with the appropriate differential equations for the physiological state Q [], the substrate(s) S [M], and the toxic product(s) P [M].
ln(K P )
ln( Y N/S )
21.1363 21.1363 16.0495 16.0492 16.0492 14.7198 14.7185 14.7185 12.8709 12.8639 12.8639 11.3036 11.2677 11.2677 9.7593 9.5507 9.5507
SSE 0.4431 0.4431 0.4431 0.3079 0.3079 0.3079 0.2865 0.2866 0.2866 0.1035 0.1034 0.1034 0.6121 0.6119 0.6119 0.6126 0.6089 0.6089
4. Back to basics: a so-called P-model as a first limiting case study of the novel class of microbial growth models 4.1. Model description In a first case study, the stationary phase is assumed to be solely resulting from toxic product inhibition. Mathematically, this implies that the factor l S (S) in Eq. (4) is equal to 1. In Table 2, some product inhibition functions reported in the literature are listed (Vereecken, 2002). In this paper the product inhibition function is assumed to be linear in the product concentration, as proposed in Ghose and Tyagi (1979). Other assump-
Table 2 Examples of product inhibition functions. P max [M]: concentration of P at which growth ceases; a [] shape parameter; k [1/M]: inhibition constant; K [M]: concentration of P for which l P ( P)=1/2 l P ( P) function
Reference
(1(( P)/( P max))) (1(( P)/( P max))a [1-((( P)/( P max)))a] exp(kd P) (((K)/(K+P)))
Ghose and Tyagi (1979) Levenspiel (1980) Luong (1985) Aiba et al. (1968) Aiba and Shocla (1969)
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tions are that (i) the initial concentration of the toxic product P(t=0) is equal to 0, (ii) there is only one growth inhibiting product, and (iii) there is only growth associated production of the toxic product P. The model consists of the following three differential equations: dN ðt Þ Q ðt Þ P ðt Þ ¼ lmax d d 1 dN ðt Þ ð5Þ dt 1 þ Q ðt Þ KP with N(t =0)= N 0 dQðt Þ ¼ lmax dQðt Þ dt
ð6Þ
with Q(t =0)= Q 0 dPðt Þ Q ðt Þ P ðt Þ ¼ YP=N dlmax d d 1 dN ðt Þ dt 1 þ Qðt Þ KP ð7Þ with P(t =0)= 0. The first equation describes the microbial evolution in time and consists of the adjustment function of the model of Baranyi and Roberts (see Eq. (1)) and an inhibition function which is linear in function of the toxic product concentration P [M]. The larger the concentration of product P(t), the smaller the increase in microorganisms. The second equation is equal to the second equation of the model of Baranyi and Roberts (Eq. (2)) and describes the exponential evolution of the physiological state. The third equation describes the evolution of the toxic product concentration P [M], with Y P/N [M/(CFU/mL)] as the yield for product over microorganisms. Dividing Eq. (5) by Eq. (7) results in dN ðt Þ 1 ¼ dPðt Þ YP=N
ð8Þ
Rearranging and integrating with the appropriate boundaries results in the following relation between N(t) and P(t): N ðt Þ ¼ N0 þ
P ðt Þ YP=N
ð9Þ
The inhibition function chosen implies that the maximum value for P is equal to K P . The asymptotic
113
microbial load N as is equal to N 0+(KP / Y P/N ). Substituting all this in Eq. (5) results in: dN ðt Þ Q N ðt Þ N0 ¼ lmax d d 1 dN ðt Þ dt 1þQ Nas N0 ð10Þ By comparing Eq. (10) with Eq. (1), it can be concluded that the P-model and the model of Baranyi and Roberts are analytically only slightly different. Since N 0 is negligible in comparison with both N(t) (for increasing time instants) and N as, both models are equal from a numerical point of view. For this model, also a static version can be derived: þKP 1 þ Q0 expðlmax tÞ N0 YP=N KP N0 ðN0 YP=N þ KP Þ 1 þ Q0 N ðt Þ¼ þKP N0 YP=N KP 1 þ Q0 expðlmax tÞ KP þ N0 YP=N 1 þ Q0 ð11Þ The factor N 0Y P/N is negligible in comparison with K P for small values of Y P/N , and by consequence, also, this static model version is analytically only slightly different from the static model version of the model of Baranyi and Roberts (see Eq. (3)). Remark that the initial microbial load and the asymptotic microbial load can also be calculated by replacing t in Eq. (11) by 0 or by l, respectively. 4.2. Model simulation A simulation study reveals the influence of the different model parameters (N 0, Q 0, l max, K P, and Y P/N ) on the microbial growth curve. The results are depicted in Fig. 4. In the upper left plot of Fig. 4, it is clearly shown that a high initial microbial load N 0 also increases the asymptotic microbial load. Although this influence is not well known (and is not present in the model of Baranyi and Roberts), it is in compliance with experimental data discussed in Carlin et al. (1995). Carlin et al. (1995) investigated the factors affecting the growth of L. innocua and reported that the population reached after 7 days (which can be considered the maximum population level) was lower for low inocula (10–103 CFU/g) compared to high inocula (105 CFU/g). This effect can be deducted by means of Eq. (9): N 0 influences the asymptotic value
114 F. Poschet et al. / International Journal of Food Microbiology 100 (2005) 107–124
Fig. 4. Influence of the different P-model parameters on the microbial growth curve. n 0=lnN 0, k=lnK P, and q 0=lnQ 0.
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of the microbial load concentration. The upper middle and right plots are similar to the corresponding ones of the model of Baranyi and Roberts. In the lower left plot, it can be seen that K P corresponds to the asymptotic microbial load for high values of K P but not for low values. This is because for small values of K P, the influence of N 0 is not negligible on the asymptotic microbial load (see also Eq. (9)). In the lower right plot, the influence of Y P /N on the asymptotic microbial load is presented. For increasing values of Y P /N , the asymptotic microbial load decreases according to Eq. (9). 4.3. Monte Carlo analysis The results of the Monte Carlo analysis (performed as described in Section 2.3) are depicted in Fig. 5. In order to have the same number of model parameters as the model of Baranyi and Roberts, the yield Y P/N is assumed to be equal to 1 (it can easily be verified that this is always possible). When comparing Figs. 2 and 5), it can be concluded that the model parameter
115
distributions, induced by experimental variation on the data, are similar for both the model of Baranyi and Roberts and this prototype P-model. 4.4. Model evaluation 4.4.1. From a P-model numerically equivalent to the Baranyi and Roberts model. . . In order to illustrate the numerical equivalence of the P-model with the model of Baranyi and Roberts, the previously mentioned L. innocua–L. lactis coculture was also described using Eq. (11) with following degrees of freedom: lnN 0, lnK P, lnQ 0, and l max. There is no visual difference between the model description of the model of Baranyi and Roberts (Fig. 3) and the P-model, and therefore, the latter is not presented in a figure; the resulting SSE and model parameter estimations are also listed in Table 1. The P-model results in equal SSE and parameter values (with N max numerically equal to N 0+K P /Y N/P ). The model of Baranyi and Roberts needs six different values of N max to describe the six different growth curves (see
Fig. 5. Model parameter distributions for the P-model using an exemplary growth curve.
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Fig. 3). This version of the P-model, described here solely for the sake of illustrating numerical equivalence with the model of Baranyi and Roberts (i.e., equal fitting properties), obviously needs six different values of K P to describe the six growth curves. 4.4.2. . . . to a more mechanistic P-model In order to encapsulate the biological mechanism for transition from the exponential phase to the stationary phase, i.e., the acidification of the medium because of the production of lactic acid, a more elaborated P-model was developed. The proton formation, which is quantitatively different for each of the six distinct growth curves, is considered as toxic product formation in this case study. The pHmeasurements (taken from Vereecken, 2002 and Vereecken et al., 2002, not shown) are used as an input for the model (i.e., the measurements are directly included in the third factor of the right-hand side of Eq. (12)). The use of the pH-measurements is equivalent to the use of the lactic acid concentrations, since for a given medium, there is a 1–1 relation between the lactic acid concentration and the pH. Mathematically, the model transformation is expressed by the following model reparameterisation: P(t)=[H +](t)=10pH(t). K P is also reparameterised to pHmin which is equal to log10(K P ). The model equations are then as follows: dN ðt Þ Qðt Þ 10pH ðtÞ ¼ lmax d d 1 pH dN ðt Þ dt 1 þ Qðt Þ 10 min ð12Þ with N(t=0)=N 0 dQðt Þ ¼ lmax d Qðt Þ dt
ð13Þ
with Q(t=0)=Q 0. Since this model incorporates the measurements of the pH-evolution (which is different for each of the six distinct growth curves and which is directly included in Eq. (12)), this model only needs a single value of pHmin (and thus of K P ) to describe all six growth curves adequately at once, whereas the model of Baranyi and Roberts needs six different values of N max to describe all six growth curves. The underlying cause of the (early induced) stationary phase, the production of lactic acid, here, modelled by the
Fig. 6. Description of the P-model on experimental data of L. innocua in different coculture experiments. The initial inoculum size of L. innocua is equal to 103 for all experiments. N 0,Ll, the initial inoculum size of L. lactis is variable: (o): monoculture, (+): N 0,Ll=103, (): N 0,Ll=104, (*): N 0,Ll=105, (D): N 0,Ll=106, (q): N 0,Ll=107.
evolution of the pH, quantitatively differs for each of the six growth curves. However, its influence on the evolution of L. innocua can be described by means of one global parameter pHmin . As a result, the total set of six growth curves is described by means of four global model parameters: ln(N 0)=7.9463 ln(CFU/mL), l max=1.5492 h1, ln( Q 0)=0.4553, and pHmin=5.6374. The biological interpretation of pHmin is clear: it is the minimum pH of growth of L. innocua. The model description on all six growth curves at once is presented in Fig. 6.
5. Back to basics: a so-called S -model as a second limiting case study of the novel class of microbial growth models 5.1. Model description In a second special case of the general model (Eq. (4)), the stationary phase is assumed to be only a result of substrate exhaustion and not of toxic product inhibition. Mathematically, this implies that the factor l P ( P) is assumed to be equal to 1. For this case study, it is also assumed that (i) a linear relation is appropriate to describe the influence of substrate depletion on the microbial growth, (ii) there is no substrate consumption for maintenance, (iii) there is
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no substrate breakdown in the medium, (iv) no additional substrate is added during the growth process, and (v) there is only one limiting substrate. These assumptions result in the following three differential equations:
Rearranging and integrating with the appropriate boundaries results in the following relation between N(t) and S(t):
dN ðt Þ Q ðt Þ ¼ lmax d dS ðt ÞdN ðt Þ dt 1 þ Q ðt Þ
The asymptotic microbial load N as is then equal to N 0+Y N/S . Substituting all this in Eq. (14) results in:
ð14Þ
N ðt Þ ¼ N0 þ YN =S d ð1 S ðt ÞÞ
ð18Þ
dN ðt Þ Q N ðt Þ N0 ¼ lmax d d 1 dN ðt Þ dt 1þQ Nas N0
with N(t =0)= N 0
ð19Þ dQðt Þ ¼ lmax d Qðt Þ dt
ð15Þ
with Q(t=0)= Q 0 dS ðt Þ Qðt Þ S ðt Þ ¼ lmax d dN ðt Þ d dt 1 þ Qðt Þ YN =S
ð16Þ
with S(t =0) =S 0. The first equation (Eq. (14)) describes the microbial evolution in time and consists of the adjustment function of the model of Baranyi and Roberts (see Eq. (1)) and an inhibition function which is, as an example, selected equal to the (rescaled) substrate concentration S. Eq. (15) is equal to the second equation of the model of Baranyi and Roberts (Eq. (2)) and describes the exponential evolution of the physiological state. The third equation (Eq. (16)) describes the evolution (consumption) of the substrate concentration S. By consequence, the right-hand side of Eq. (16) is, except for the minus sign and the yield coefficient Y N/S , equal to the right-hand side of Eq. (14). The substrate concentration is rescaled in order to obtain a strictly monotone decreasing inhibition function with values between 1 and 0 and to have a dimensionless substrate concentration. Alternative substrate inhibition functions, inspired on, e.g., Monod kinetics, would obviously also be possible (with other assumptions). Dividing the first model equation (Eq. (14)) by the third one (Eq. (16)) results in dN ðt Þ ¼ YN =S dS ðt Þ
ð17Þ
Eq. (19) is equal to Eq. (10), and after comparison with Eq. (1), it can be concluded that also the S-model and the model of Baranyi and Roberts are analytically only slightly different. Since N 0 is negligible in comparison with both N(t) (for increasing time instants) and N as, both models are equal from a numerical point of view. For this model, a static version can be derived:
N0 þYN =S 1 þ Q0 expðlmax tÞ YN =S N0 ðN0 þ YN =S Þ 1 þ Q0 N ðt Þ ¼ N0 þYN=S 1 þ Q0 expðlmax tÞ YN =S YN =S þ N0 1 þ Q0 ð20Þ Again, this static model version analytically slightly differs from the static model version of the model of Baranyi and Roberts. Since N 0 is negligible compared to Y N/S , both models can be considered as numerically equal. The initial microbial load N(t=0), which is calculated by replacing t in Eq. (20) by 0, is equal to N 0. 5.2. Model simulation A simulation study reveals the influence of the different model parameters (N 0, Q 0, l max, Y N/S , and S 0) on the microbial growth curve. The results are depicted in Fig. 7. In the upper left plot of Fig. 7, it is clearly shown that a high initial microbial load N 0 increases the asymptotic microbial load. Again, this is in compliance with experimental the data presented in Carlin et al. (1995) and can be deducted by means of Eq. (18): N 0 influences the
118 F. Poschet et al. / International Journal of Food Microbiology 100 (2005) 107–124
Fig. 7. Influence of the different S-model parameters on the microbial growth curve. n 0=lnN 0, y=lnY N/S , and q 0=lnQ 0.
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(asymptotic) value of the microbial load concentration. The upper middle and right plots are similar to the corresponding ones of the model of Baranyi and Roberts. In the lower left plot, it can be seen that Y N/S corresponds to the asymptotic microbial load for high values of Y N/S but not for low values. This is also explained by interpretation of Eq. (18): for a high value of Y N/S , the contribution of N 0 to the (asymptotic) value of the microbial load concentration is negligible. The lower right plot illustrates the influence of S 0 on the asymptotic microbial load, also experimentally reported in Leroy et al. (2003) who investigated the influence of the initial glucose concentration, amongst others, on the maximum biomass concentration of Enterococcus faecium RZS C5. S 0 also influences the maximum specific growth rate. Both influences are also reported for exciting bacteria in Verluyten et al. (2002). The influence of the initial substrate concentration S 0 on the maximum microbial load population is also depicted in the left part of Fig. 5, Van Impe et al. (2004).
119
5.3. Monte Carlo analysis The results of the Monte Carlo analysis (performed as described in Section 2.3) are depicted in Fig. 8. In order to have four model parameters as in the model of Baranyi and Roberts, it is assumed that S(t =0) = S 0 is equal to 1. A comparison of Figs. 2 and 8 reveals that the model parameter distributions are similar for the model of Baranyi and Roberts and the S-model. 5.4. Model evaluation Although substrate S is not the limiting factor in this case, the separate growth curves of the previously mentioned L. innocua–L. lactis coculture can also be described with Eq. (20) to illustrate the equal fitting capacity of the S-model vs. the model of Baranyi and Roberts. There is no visual difference between the model description of the model of Baranyi and Roberts (Fig. 3) and the S-model, and therefore, the latter is not presented in a figure; SSE and model
Fig. 8. Model parameter distributions for the S-model using an exemplary growth curve.
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parameter estimations are also listed in Table 1. The S-model results in equal SSE and parameter values (with N max numerically equal to N 0+Y N/S ). Selfevidently, it is not claimed that this ad hoc procedure would be more suitable than the model of Baranyi and Roberts, and for case studies where substrate is a limiting factor, (advanced) substrate measurements are mandatory for a sound application of these S-type models. An example of such an S-type model is presented by Leroy and De Vuyst (2001). In that research, the growth inhibition was represented as a combined effect of lactic acid inhibition, sugar limitation, and nutrient depletion.
concentration for organism i. is selected equal to 10–6. This small value is needed from a numerical point of view to facilitate the estimation of parameters [LaH]max,i,G and [H +]max,i,G . The above equation, introduced in Vereecken et al. (2003), describes the growth suppression by both pH and undissociated lactic acid on both organisms, with the target being the more sensitive one. A second (biological) part of the model describes the production of lactic acid by both organisms:
6. A more elaborated P-model for microbial interaction
mLaH;i ðLaH Þ ¼ 1
In this final section, a more realistic (and more complex) P-model describing the interaction between a target organism, L. innocua, and a microbial antagonist, the lactic acid bacterium L. lactis, is presented. The inhibition of the target organism is explicitly related to the lactic acid production curve. Both inhibitory effects of lactic acid—the lowering of the medium pH and the formation of undissociated lactic acid (as mentioned above)—are included in the model. This model can by consequence be seen as a further extension of the model presented in Section 4.4.2. The model originates from Vereecken (2002). A first (biological) part of the model considers the growth of target (i=1) and antagonist (i=2): dNi ðt Þ Qi ðt Þ ¼ li dNi ðt Þ ¼ lmax;id dlLaH;i dNi ðt Þ dt 1 þ Qi ðt Þ
lLaH;i
¼ 1
½LaH ½LaH max;i;G
1þ d 1
½H þ ½H þ max;i;G
1þ
ð21Þ with N i (t=0)=N 0,i with [H +] [M] the proton concentration, [LaH] [M] the undissociated lactic acid concentration, [LaH]max,i,G [M] and [H +]max,i,G [M] the growth inhibiting lactic acid concentration and the growth inhibiting proton
2 X dLaH tot ¼ ðYLaH=Ni dli þ mmax;i dmLaH;i ðLaH ÞÞdNi dt i¼1
½LaH ½LaH max;i;M
1
1þ
½H þ ½H þ max;i;M
1þ ð22Þ
with LaHtot [M] the total lactic acid concentration, Y LaH/N i [mmol/CFU] the yield of lactic acid for organism i, m max,i [mmol/(CFU h)] the maintenance coefficient of organism i, and [LaH]max,i,M [M] and [H +]max,i,M [M] the metabolism inhibiting lactic acid concentration and the metabolism inhibiting proton concentration for organism i. This equation is proposed in Vereecken et al. (2002). The last (chemical) part concerns the dissociation kinetics of lactic acid in the medium, for which the buffering capacities are not exactly known. [LaH] and pH are computed in accordance with the method described in Vereecken and Van Impe (2002). In order to represent the relation between [LaH] and LaHtot , the following relation, inspired by Dabeskinetics, is proposed. ab ½LaH ¼ aLaH tot ðLaH tot þ bÞ 2ð b c Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðLaH tot þ bÞ2 4ðb cÞLaH tot ð23Þ An interpretation of the model parameters a, b, and c is presented in Vereecken and Van Impe (2002). To
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Fig. 9. Experimental data and model description (Eqs. (20)–(23)) of the L. lactis–L. innocua coculture. (o): cell concentration of the antagonist, (w): cell concentration of the target (: data point of the target below detection limit), (D): lactic acid concentration, (*): pH. Dash-dotted line: simulation beyond experimentally observed stationary phase.
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represent the relation between pH and [LaH], a second expression, also based on the Dabes-kinetics, is developed as follows. pH ¼
1 ½ðb 2c1 Þ½LaH 2a1 c1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b21 ½LaH2 þ 4a1 b21 c1 ½LaH þ pH 0
ð24Þ
Model parameters a 1, b 1, and c 1 have a similar interpretation as a, b, and c, respectively, and which can also be found in Vereecken and Van Impe (2002). pH0 symbolises the initial pH. Eqs. (23) and (24) determine the medium-related parameters. The procedure should be accomplished once for a specific medium and initial pH. Indeed, since the model parameters a, b, c, a 1, b 1, and c 1 only depend on the medium characteristics and pH0 only on the initial pH, they are suited for every growth experiment in this particular medium and initial pH. After calibration, prediction of the time dependent course of pH (and [LaH]) is always possible when a only LaHtot profile is measured. Application of the complete model to the experimental data set of cell concentrations N i , total lactic acid concentration LaHtot, and pH consists out of two major steps: (i) determination of the medium related parameters (i.e., the parameters of Eqs. (23) and (24)) and (ii) determination of growth and production related parameters (i.e., the parameters of Eqs. (21) and (22)). In the first step, the [LaHtot(t), pH(t)] measurements are the input to estimate the model parameters of Eqs. (23) and (24), without considering the cell concentration data. To identify the parameters of Eqs. (21) and (22), describing the cell growth and metabolite production, the complete data set (including cell counts, lactic acid, and pH measurements) is necessary. Eqs. (23) and (24) are used to determine the input values [LaH](t) and [H +](t) of the specific growth and production rates (Eqs. (21) and (22)). Fig. 9 represents the experimental data points of cell concentration, total lactic acid production and pH, and the highly accurate description with the model (Eqs. (21)–(24)) for three experiments of the L. innocua–L. lactis coculture at 35 8C, namely, the monocultures and one coculture (with a L. lactis inoculum level of 105 CFU/ mL). The model contains 14 model parameters: four
parameters of growth, three parameters for lactic acid production, and seven parameters for the lactic acid dissociation. More details can be found in Vereecken (2002). For each individual growth (monoculture/ coculture) curve, the onset of the stationary phase coincides with the large rate of increase of lactic acid production. In the coculture experiments, two antagonistic effects emerge, namely, the early induction of the stationary of the target and a decline phase, where the target’s cell concentration is reduced to below detection limit. Note that, in view of the model equations, only the inhibiting effect of lactic acid is considered (i.e., the induction of the stationary phase) and not the inactivation effect (explaining the decline phase).
7. Conclusions The main contribution of this paper is the analysis of a novel class of predictive microbial growth models which reflect the (micro)biological phenomena governing the microbial growth process. This research particularly focuses on the transition from the exponential growth phase to the stationary phase, which is generally assumed to be the result of substrate depletion and/or toxic product inhibition. Two limiting case studies of the novel class of microbial growth models are carefully analysed and compared to the model of Baranyi and Roberts, the currently most used logistic-type model. Both the socalled P-model and S-model (i) have an equal fitting capacity as the model of Baranyi and Roberts, (ii) have identical model parameter distributions, (iii) are easier to extend to more complex situations, and (iv) are applicable to both the macroscopic (i.e., population) and the microscopic (i.e., individual cell) levels. A more complex P-model describing the interaction between a target organism and an antagonist through the formation of lactic acid by the antagonist is presented to illustrate the power of the novel class of models.
8. Uncited references Bailey and Ollis, 1986 Ciftci et al., 1983
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Foegeding, 1997 ICMSF, 1988 McMeekin et al., 1992 McMeekin et al., 1997 Motarjemi and KRaferstein, 1999 Ross, 1999 Zwietering and van Gerwen, 2000
Acknowledgements This research is supported by the Research Council of the Katholieke Universiteit Leuven as part of project IDO/00/008, the Institute for the Promotion of Innovation by Science and Technology (IWT), the Fund for Scientific Research– Flanders (FWO) as part of project G.0213.02 and for the Postdoctoral Fellowship of AG, the Belgian Program on Interuniversity Poles of Attraction and the Second Multiannual Scientific Support Plan for a Sustainable Development Policy, initiated by the Belgian Federal Science Policy Office, and the European Commission as part of project QLK1-CT2001-01415. The scientific responsibility is assumed by its authors.
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