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), the physiological state of the cells (), substrate concentrations () and other factors. The presence and absence of the factors of each category depends on their relevance in the microbial process under study. Microbial growth is obtained when µ i > 0 and microbial decay when µ i < 0. The development of interaction models with a (semi-)mechanistic basis is usually based on the incorporation of growth-influencing effects, that can be ascribed to all competing microbial populations, into the individual growth kinetics of each single population in equation (11.1) (Breidt and Fleming, 1998; Martens et al., 1999; Leroy and De Vuyst, 2003, 2005; Leroy et al., 2005a,b; Poschet et al., 2005). Examples of such growth-influencing effects are changes in pH and in the concentration of nutrients and inhibitory compounds such as organic acids. Their quantification is a way of incorporating environmental changes due to the action of one or more competing microbial populations into the model (Malakar et al., 1999; Poschet et al., 2005).

11.3.2 Developing a model to describe interactions that cause growth inhibition Growth inhibition of a given microbial population can be ascribed to selfinhibition (Leroy and De Vuyst, 2001), for instance due to its own depletion of nutrients, as well as to inhibitory effects caused by competing populations (Fig. 11.2). Nutrient depletion is generally not of concern in foods that are not subject to a high degree of spoilage, except for some fermented food products such as fermented sausage, where the depletion of sugar or, possibly, of growthstimulating peptides may be of importance. If one is dealing specifically with a microbial interaction situation that causes growth inhibition, the quantification of the inhibitory effect γx of a population Ni, due to a specific effect x caused by one or more other populations, can be written symbolically as the ratio of the actual value of the specific growth rate (µi) to its optimum value in the absence of the

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Fig. 11.2

Inhibitory effects on the evolution rate of a microbial population in a food ecosystem containing competing microbial populations.

growth-limiting effect, i.e. at the optimum value of x (µi opt(x)), provided that all other factors remain constant: γx = µi /µi opt(x)

[11.2]

where γx for a studied effect ranges from one (no inhibition) to zero (complete inhibition). Such changes in specific growth rate are usually of a dynamic nature, thus changing over time. For instance, the growth inhibition of a certain bacterial population can be related to a continuous decrease in pH due to acidification by one or more other populations. This can for instance be quantified as (Rosso et al., 1995; Malakar et al., 1999): (pH – pHmin)(pH – pHmax) γpH = ––––––––––––––––––––––––––––––––– (pH – pHmin)(pH – pHmax) – (pH – pHopt)2

[11.3]

The latter equation is a cardinal function based on the maximum, minimum and optimum values of pH for growth to be estimated in the absence of organic acids. Besides the pH effect, specific inhibition by undissociated organic acid molecules should be considered [see below, equation (11.6)]. Further, growth inhibition can be due to the exhaustion of a certain nutrient or substrate S (e.g. glucose in grams or moles per unit of volume) by a competing population. This effect can be described with Monod-type kinetics (Malakar et al., 1999): γ[S] = S/(KS + S)

[11.4]

with KS the Monod constant (in grams or moles per unit of volume). If the substrate

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is a sugar, the changes in S can be calculated with Pirt-type equations that take into account sugar consumption for biomass production as well as for cell maintenance and product formation, for all relevant populations (Pirt, 1975; Malakar et al., 1999). A similar approach is followed in the so-called S-model by Poschet et al. (2005). However, such ‘smooth’ and somewhat simplistic exhaustion kinetics are not always satisfactory. Nutrient depletion sometimes follows more complex and less smooth patterns, especially for lactic acid bacteria that have high and complex nutrient requirements. In such cases, the nutrient depletion model offers an alternative (Leroy and De Vuyst, 2001). γ[CNS] = 1

[X] ≤ [X1]

γ[CNS] = 1 – I1 ([X] – [X1])

[X1] < [X] ≤ [X2]

γ[CNS] = (1 – I1 ([X2] – [X1]) – I2 ([X] – [X2])

[X] > [X2]

[11.5]

with [X1] and [X2] the critical cell densities for nutrient inhibition (in cfu l–1) and I1 and I2 the dimensionless inhibition slopes. The values of [X1], [X2], I1, and I2 are expected to be a function of the competing microbial populations. Finally, the effect of growth inhibitory metabolites can be quantified. For instance, the inhibition due to the presence of an organic acid A (e.g. lactic acid) is related to the concentration of undissociated organic acid ([HA], in grams or moles per unit of volume) as follows (Breidt and Fleming, 1998; Leroy and De Vuyst, 2001):  [HA]  γ[A] = 1 – –––––––  [HA]max 

n

[11.6]

with n a fitting exponent and [HA]max the maximum value of [HA] that still allows growth. The increase in [A], and the resulting, pH-dependent [HA], can be calculated based on growth- and non-growth-related terms of the acid-producing population(s) (Luedeking and Piret, 1959; Malakar et al., 1999). This accumulation of undissociated organic acid as a specific antimicrobial adds to the effect of the decrease in pH caused by acidification [see above, equation (11.3)]. Undissociated organic acid molecules are uncharged and may therefore cross the cell membrane and acidify the cytoplasm, resulting in cell death. A similar approach for product inhibition is followed in the P-model applied by Poschet et al. (2005). It is important to realize that growth inhibition due to microbial interactions is seldom the result of one single growth inhibitory action, but usually of the combination of several effects. Interestingly, the combined effects of different inhibitory actions can sometimes be obtained by simply considering such effects as multiplicative. According to the γ-concept, overall growth inhibition can be obtained by multiplying the individual γ-functions, including the ones listed above (te Giffel and Zwietering, 1999; Leroy and De Vuyst, 2001). If this is not the case, for instance at the growth/no growth interface, more complicated and interdependent quantification methods have to be developed.

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11.3.3 Developing descriptive interaction models A mechanistic approach requires a complex mathematical model with many variables. Therefore, less complex descriptive interaction models are built by simply quantifying how much the growth of a population is affected by the growth of one or more other populations. Therefore, the growth kinetics of pure cultures are usually compared with the growth kinetics obtained in mixed cultures. Statistical tests (e.g. F-test) are then used to evaluate if the interaction effects are significant (Pin and Baranyi, 1998). Also, the growth of two mixed populations Ni and Nj (in number of cells per unit of volume) can be described with a relatively simple Lotka-Volterra-type model for two-species competition for a limited amount of resources, as follows (Dens et al., 1999; Vereecken et al., 2000; Dens and Van Impe, 2001). Qi(t) 1 µi = µmax ––––––– –––– (Nmax – Ni – αij Nj) i 1 + Q (t) i Nmax i

[11.7]

i

with Nmaxi the maximum population density of species i when no other species is present, Qi(t) the physiological state of the cells needed to describe the lag phase and αij a coefficient of interaction measuring the effects of species j on species i.. As is the case for the specific growth rate, it is conceivable that the lag phase could also be affected by microbial interactions. However, conclusive research on this topic is currently lacking.

11.3.4 Developing interaction models based on antimicrobial activity An extension of the model presented above incorporates the antagonistic effects due to the production of antimicrobials (e.g. bacteriocins) by population Ni on a sensitive population Nj, as follows (Pleasants et al., 2001; Leroy et al., 2005a,b): Qj(t) 1 µj = µmax ––––––– –––– (Nmax – Nj – αji Ni) – ψB j 1 + Q (t) j Nmax j

[11.8]

j

with ψ the bacteriocidal coefficient (in units of volume per hour per activity unit) and B the concentration of antimicrobial (in activity units per volume). Figure 11.3 gives an illustration of the inactivation of a bacteriocin-sensitive population of L. innocua LMG 13568 in a mixed culture with the bacteriocin-producing strain Lb. sakei CTC 494 (Leroy et al., 2005b). The data are modelled with equation (11.8). A quick drop in Listeria counts is observed from the very moment bacteriocin is being detected. A bacteriocin-resistant subpopulation of L. innocua LMG 13568 is obtained, but is not able to grow out because of the stringent environmental conditions. In such cases, it is convenient to split the target population into a bacteriocin-resistant and a bacteriocin-sensitive subpopulation. In monoculture, L. innocua LMG 13568 does not display an inactivation pattern.

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Fig. 11.3 Evolution of Listeria innocua LMG 13568 counts (in cfu ml–1) in MRS medium at 20 °C and a constant pH of 5.2, containing 40 g l–1 of sodium chloride, and 200 ppm of sodium nitrite, in monoculture (õ) and in mixed culture (¤) with the bacteriocin-producing strain Lactobacillus sakei CTC 494 (◊). Bacteriocin activity by Lb. sakei CTC 494 is presented in Arbitrary Units (AU) per ml (˜) and detection limits inherent to the associated bioassay are given by the bars. Full lines are according to the interaction model by Leroy et al. (2005a,b). The dashed line indicates the onset of bacteriocin production resulting in a decrease of Listeria counts.

11.4 Applications and implications for food processors 11.4.1 Improvement of food safety and shelf stability Modelling of microbial processes in foods, in particular predictive food microbiology, yields quantitative information that is crucial for process control, food safety and the prevention of food spoilage (McDonald and Sun, 1999). Validated predictive models represent an essential tool to aid the exposure assessment phase of ‘quantitative risk assessment’ (Ross et al., 2000). In the framework of quantitative risk assessment, models for microbial interaction should not be neglected, because the behaviour of foodborne pathogens may be strongly affected by the growth of competing microbiota, in particular in fermented products or if competing differences are relatively large.

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11.4.2 Fermented foods Fermented foods, characterized by strong antimicrobial production and pronounced microbial interactions, represent an interesting field of application for models that describe microbial interactions. Lactic acid bacteria, yeasts and fungi play an important role in the production of fermented foods. In such foods with high microbial loads, lactic acid bacteria generally display complex patterns of inhibitory actions towards other microbial populations, including thorough acidification and the production of several antimicrobial compounds. As a result, lactic acid bacteria populations are generally highly competitive in food ecosystems, provided that they are adapted to the substrate (Leroy et al., 2002). Based on their antagonistic activities, the use of lactic acid bacteria as functional starter cultures in fermented foods or as bioprotective cultures in non-fermented foods is a way to stabilize the food ecology (De Vuyst, 2000; Leroy and De Vuyst, 2004). Nevertheless, for this strategy to be successful, it is essential to understand the mechanisms that lead to the competitive advantages of lactic acid bacteria and to quantify their interaction effects with spoilage bacteria and foodborne pathogens in food systems (Breidt and Fleming, 1998). Studying such interactions can help to optimize process technology, because it can reveal and quantify crucial elements such as inactivation limitations due to bacteriocin-resistant subpopulations (Leroy et al., 2005a,b).

11.4.3 Product quality and attractiveness Applications should not only focus on antagonisms. In situations where interactions are of a positive nature, for instance during protocooperation by yoghurt cultures (Courtin et al., 2002; Ginovart et al., 2002), beneficial effects such as enhanced acidification and aroma formation may be obtained compared to pure cultures. An optimization of the beneficial effects is to be obtained by steering the interactions towards improved performance of the interacting microbial populations.

11.5 Future trends 11.5.1 Improvement of the model architecture While models are very useful decision support tools, they remain simplified representations of reality. Therefore, model predictions should be used with cognisance of microbial ecology principles that may not be included in the model, and efforts should be made to incorporate such missing principles (Ross et al., 2000). Microbial risk assessment processes, for instance, are evolving to incorporate more science to replace judgements that do not hold up to hypothesis testing in controlled scientific experiments (Coleman et al., 2003). This certainly leaves room for improvement of the interaction models. Future trends will probably deal with the improved extrapolation of laboratory studies in liquid media to real food environments. Growth models based on experiments in broth

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frequently overpredict growth in the actual food environment (Houtsma et al., 1996; Ross et al., 2000; Coleman et al., 2003). One of the major reasons is probably because food is often solid or semi-solid and spatially heterogeneous. In such structured, (semi-)solid food matrices, microorganisms generally grow as immobilized colonies (Wilson et al., 2002). In fermented sausage, for instance, the distance between ‘nests’ of lactobacilli varies from 100 to 5000 µm (Katsaras and Leistner, 1988, 1991). As a result, the spatial separation of the colonies and the gradients caused by diffusion limitations (e.g. pH, substrate, oxygen, antimicrobials) will influence potential interactions between them. Indeed, spatial differentiation has an influence on the behaviour (coexistence/extinction) of the populations (Dens and Van Impe, 2001). This opens the way for the development of spatial or territory models. In such models, the statistical geometric distribution of the colonies, the local exhaustion and diffusion of nutrients, and the local accumulation and diffusion of inhibitory compounds have to be taken into account (Thomas et al., 1997; Dens and Van Impe, 2001).

11.5.2 Modelling of positive interactions The study of positive interactions instead of negative, inhibitory-type interactions seems promising. In many foods, food quality and safety rely heavily on the presence of distinct beneficial populations. In yoghurt, for instance, the interplay of formic acid, peptides, and acidity between S. thermophilus and Lb. delbrueckii subsp. bulgaricus should be modelled to understand bacterial growth kinetics leading to protocooperation (Ginovart et al., 2002). Likewise, modelling can be applied to study the remarkably stable associations between certain yeasts and lactic acid bacteria observed in sourdough (Gänzle et al., 1998). In fermented sausage, certain lactic acid bacteria, catalase-positive cocci, yeasts, and fungi all play a role, and it is evident that a certain degree of interaction between the microbial groups will occur, in particular regarding flavour development (Sunesen et al., 2004). Interaction studies should therefore not only focus on growth and inactivation of microorganisms, but also try to quantify other aspects such as the production of aroma compounds or molecules that are advantageous to the health of the consumer. Moreover, little quantitative information is available about the stimulatory effects of proteolysis of food proteins by certain microbial populations on the growth of other populations, despite the important role of hydrolysis of proteins in microbial growth kinetics (de la Broise et al., 1998).

11.5.3 Application of interaction models in population dynamics and ecology Based on the increased knowledge on heterogeneity within microbial strains, interactions between different subpopulations of a microorganism may be considered. Consideration of the fitness costs of certain specific mutations and their advantages in microbial interactions may help to predict the prevalence and stability of a mutant within a certain population. For instance, the occurrence of

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bacteriocin-resistant subpopulations of Listeria on strain level may be the result of repeated exposure to bacteriocin-producing lactic acid bacteria. Also, in certain cases, the effects of cell-to-cell communication and quorum sensing should be explored when considering microbial interactions and growth kinetics (Kaprelyants and Kell, 1996). This may for instance help to explain the ecological phenomenon where growth inhibition of slow-growing microorganisms is related to the total density of all the microbial populations present in the food (Coleman et al., 2003). Such interaction models would have to be of a mechanistic nature and may be based on molecular biology data.

11.5.4 Integrating interaction models in risk assessment Finally, integration of microbial interaction models in risk assessment is a challenge for the future. However, one should bear in mind that interactions between microbial populations are part of a highly complex process and that any reduction of an ecosystem to a few populations, or inappropriate selection of strains, involves considerable risks (Coleman et al., 2003). For instance, the level of antagonism in the inhibition of Salmonella Typhimurium in the presence of five representative species of indigenous competitors of the gut microbiota in a continuous culture broth system was reduced if any one of the competitors were removed from the culture (Ushijima and Seto, 1991). Such results caution against oversimplifications of complex ecosystems. Therefore, targeted research will be needed to calibrate adjustments of available predictive models for risk assessment with the dense and diverse populations of food ecosystems (Coleman et al., 2003).

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