Gamma Study of pH, Nitrite, and Salt Inhibition of Aeromonas hydrophila

The gamma hypothesis states that there are no interactions betweenantimicrobial environmental factors. The time to growth of Aeromonashydrophila challenged with pH, NaNO₂, and salt combinationsat 30°C was investigated. Data were examined using a modelbased on the gamma hypothesis (the gamma model), which takesinto account variance-stabilizing transformations and whichgives biologically relevant parameters. At high concentrationsof NaNO₂ and at pHs of >6.0, the antimicrobial action ofthe nitrite ion has a strong influence (MIC = 2,033 mg liter^–1),whereas at pHs of <6, nitrous acid is dominant (MIC = 1.5mg liter^–1). This change is not due to a "synergy" betweenpH and the nitrite ion but is due to the shift in the equilibriumconcentrations of nitrous acid and nitrite in solution causedby pH. In combination with salt, the parameters found for theaction of Na nitrite were identical to those found when it wasexamined in isolation. Therefore, pH, NaNO₂, and salt act independentlyon the growth of A. hydrophila. By expanding the gamma modelwith a cardinal temperature model, the results of fitting themodel of Palumbo et al. (J. Food Prot. 54:429-435, 1994) torandomly produced environmental conditions could be reproduced,suggesting that temperature also has an independent effect.

INTRODUCTION

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Introduction

Antimicrobials in combination can give an advantage over oneused alone. For example, the interaction of factors such assalt, pH, and sodium nitrite with low temperature inhibits awide range of food-borne pathogens, including Salmonella, Listeriamonocytogenes, Aeromonas hydrophila, and Clostridium botulinum(12, 7, 29, 14). In the last case, such combinations can assurethe safety of sous-vide foods (13).

The hurdle concept developed by Leistner (19) can be definedas a "systems approach" to food preservation. The concept considersthe aggregation of various preservation processes—chemical,physical and biological—to control the growth of spoilageor pathogenic organisms in foods. The hypothetical basis ofhurdle technology is that the combination of several inhibitoryprocesses or events (hurdles) gives a multitarget disturbanceof homeostasis (20, 21). Combinations are considered, therefore,to achieve results better than or at least equivalent to thoseof a single inhibitory action. Within hurdle technology, theidea that combined inhibitory factors can lead to "synergy"is an oft-vaunted hypothesis. Brocklehurst (5) states, however,that although hurdles such as pH, temperature, and water activity(a_w) act independently, "It would be expected... that interactionsmust occur between certain hurdles." The interaction betweenweak acids and pH and the interaction terms from polynomialpredictive models are used as confirmatory examples.

Predictive microbiology or "the quantitative microbial ecologyof foods" (25, 26) attempts to provide mathematical models ofmicrobial growth under a variety of environmental conditions—e.g.,temperature, pH, a_w, and the effect of preservatives. Predictivemodeling can be considered the quantification of hurdle technology.The variety of models, of modeling procedures, and of data typesand the intrinsic variability observed within the microbialresponses show predictive modeling to be an active, developingfield of study (24) and therefore a field that also has activedebate on definitions and concepts in use.

Ratkowsky (32) heads a section in a chapter on modeling with"Why polynomial models do not work." Although the main argumentwas that such models were not parsimonious, that they lackedphysically interpretable parameters was also a major concern.Polynomials do work, however, in the sense that they are fit-for-purpose,the purpose being to empirically describe the observed data,enabling the interpolation of the data set. The nontheoreticalbasis of such response-surface models was previously statedby Gibson et al. (12) in her landmark publication on Salmonella.Others have suggested, however, that the cross terms of polynomialequations have a physical significance by attributing statisticallysignificant cross terms to interactive effects between factors(e.g., see references 4, 5, and 7).

The gamma concept (36) suggests that combined environmentalfactors (temperature, pH, a_w, etc.) independently affect thegrowth of microorganisms. More recently, this fundamental concepthas been abandoned by some authors, who have assumed that theoriginal concept can be enlarged by the addition of arbitraryinteractive effects, e.g., interactions between weak acids andpH (22) or between weak acids (10), as well as a study of theinteractive effects between temperature, pH, and a_w (1), whichcounters Brocklehurst (5).

The growth of A. hydrophila in combinations of temperature,pH, salt, and NaNO₂ has been reported (29, 30), and the polynomialobtained forms the basis of the predictive model used in thePathogen Modeling Program (U.S. Department of Agriculture).The apparent interactions between pH, nitrite, and salt havealso been commented on for other organisms (3, 4, 6, 8, 33).The nitrite ion is in equilibrium with nitrous acid (HNO₂; pKa= 3.38), and the antimicrobial effect of nitrite is associatedwith the activity of nitrous acid (9) and as such is intimatelylinked to the environmental pH.

Recently we showed that combinations of pH, salt, and some commonweak acid preservatives act independently on the growth of A.hydrophila as monitored by optical density (OD) (16). Giventhe importance of combinations of pH, salt, and nitrite withinthe literature and whether interactions between these factorsexist or are an artifact of the models used, it appeared appropriateto develop our study of this organism using these particularcombinations.

MATERIALS AND METHODS

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Materials and Methods

Bacteria and culture conditions.
Aeromonas hydrophila (ATCC 7966) was grown overnight in a flaskcontaining 80 ml tryptone soya broth (Oxoid CM 129) with shakingat 30°C. The cells were harvested, centrifuged to a pellet,washed, and resuspended in peptone (0.1%). A standard inoculumwas produced by diluting the culture to an OD of 0.5 at 600nm. This standardized culture was then further diluted to producethe starting inoculum of approximately 1 x 10⁵ CFU ml^–1.

Inhibitor analysis.
All analyses were performed by using a Bioscreen microbiologicalanalyzer (Labsystems, Helsinki, Finland) incubated at 30°C.The inhibitory effect of NaNO₂ and pH was analyzed by usingthe general method described by Lambert and Pearson (18), withreplicate plates of NaNO₂ concentrations made up at differentpH values (total of 495 different conditions). For the analysisof salt and NaNO₂ at various pHs, checkerboards of the combinedinhibitors were prepared at a constant pH, using the methodof Lambert and Lambert (17). A total of 495 different conditionswere analyzed.

The Bioscreens were set to take an OD reading at 600 nm every10 min.

Model fitting.
Data obtained from the Bioscreen (tables of OD and time) weretransformed to either the reciprocal or natural logarithm. Suchtransformations stabilize the data variance, allowing a less-biasedanalysis. Wells that showed no growth during the period of theexperiment were not used in the model fitting (censored) butwere retained for comparison and also for the logistic modelingof the growth/no-growth boundary.

The general form of the model used in these studies was developedfrom that introduced by Lambert and Lambert (17), which wasused to examine data from checkerboard experiments with combinedantibiotics and other microbial inhibitors. The model is capableof handling the required variance-stabilizing transformationsnecessary to ensure the minimization of bias through regressionanalysis (35, 32):

Formula 1 1
whereTTD_ref and TTD_obs are the reference time to detection (normallytaken to be the shortest time to detection) and the observedtime to detection for a given set of environmental conditions,and q is the power transformation of the TTD data which givesapproximately equal variance. The summation term gives the functionfor the inhibitory effect of n inhibitors, each of which isdefined by two parameters, P₂_i _– ₁ and P₂_i: the parameterP₂_i _– ₁ is the concentration of the inhibitor which givesan inhibition of growth relative to the optimal of 1/e (approximately0.368), and the exponent parameters (P₂_i) are measures of theslope and can be considered a measure of the dose response.

Since q is rarely known, two principal values of q were assumed(q = 2 or 4), corresponding to the log and reciprocal transformations(termed the log model and reciprocal model, respectively).

MICs were calculated using equation 2, following the methoddescribed by Lambert and Lambert (17).

Formula 2 (2)
The concentration of weak acid at a specificpH was calculated using the Henderson-Hasselbalch equation (equation3).

Formula 3 (3)
The equationwas fitted to data using nonlinear regression with the minimizedsum of squares as the search criterion. Analyses were done usingthe JMP Statistical Software (SAS Institute, Cary, NC).

Logistic modeling of the growth/no growth boundary was doneby degrading the data to nominal data and using the nominallogistic modeling application within JMP. Response surface fitsfor comparison to those obtained from equation 1 were also performedwith the JMP software. The scaling factors given are those calculatedwith the JMP software.

RESULTS

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Results

Nitrite and pH.
The OD/incubation time profiles for the inhibition of A. hydrophilachanged with changing pH. At pH 6.8, a decreasing maximum ODwith increasing total NaNO₂ concentration was observed, whereasat pH 5.5, identical OD/incubation time profiles were observedbut were displaced down the time axis with increasing NaNO₂concentrations. By plotting the integral of the OD curve (areaunder the curve) against the incubation time (Fig. 1a and b),the differences between the two types can be seen. The gradientof each curve is the maximum OD attained. At pH 6.8, this isdecreasing with the NaNO₂ concentration, whereas at pH 5.5,this is constant.

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FIG. 1. (a) Plot of the incubation time against the area under the OD/incubation time curve for several concentrations of sodium nitrite at pH 6.8. (b) Plot of the incubation time against the area under the OD/incubation time curve for several concentrations of sodium nitrite at pH 5.5.

The gamma model (equation 1) was constructed with three possibleantimicrobial effects: pH, the nitrous acid concentration, andthe nitrite concentration (calculated from equation 3),

Formula 4 (4)
At a specific pH, the inhibitoryfunction for pH is assumed to be independent of the other antimicrobialcomponents (assertion of the gamma hypothesis) and is thereforeconstant, and equation 4 can be rewritten as

Formula 5 (5)
where RRD_pH is the relative rate to detectionat a constant pH. Since at a constant pH the ratio of nitrousacid to nitrite ion is also constant (equation 3), the right-handside can also be replaced, to a first approximation, by

Formula 6 (6)
where the parameters C_i representcombined parameters.

Fitting equation 6 to the iso-pH data gave the parameters listedin Table 1; MICs were calculated using equation 2. A plot ofpH against MIC (mg liter^–1 of total NaNO₂) gave a goodfit to a simple quadratic: MIC_{total nitrite} = 812.3 pH² –8,502.5 pH + 22,930; r² = 0.999. From the MIC and the pH, theconcentration of nitrous acid ranges between 0.8 mg liter^–1at pH 6.8 and 1.6 mg liter^–1 at pH 5.7. These values aresimilar to those found by Castellani and Niven with Staphylococcusaureus (9). Figures 2 and 3 show the fit of equation 6 and itslogarithmic transform, respectively, to the iso-pH data.

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TABLE 1. Sodium nitrite inhibition of A. hydrophila^a

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FIG. 2. Effect of total nitrite and pH on RRD: pHs 6.8 ( ${blacksquare}$ ), 6.5 ( ${square}$ ), 6.2 ( ${blacklozenge}$ ), 5.9 ( ${diamond}$ ), 5.7 ( ${blacktriangleup}$ ), and 5.5 ( ${triangleup}$ ).

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FIG. 3. Effect of total nitrite and pH on ln(TTD): pHs 6.8 ( ${blacksquare}$ ), 6.5 ( ${square}$ ), 6.2 ( ${blacklozenge}$ ), 5.9 ( ${diamond}$ ), 5.7 ( ${blacktriangleup}$ ), and 5.5 ( ${triangleup}$ ).

When the full data set was examined using equation 4 and a comparisonmade with a model for pH and nitrous acid as the sole inhibitoryfactors, the conclusion reached was that the nitrite ion hada significant antimicrobial action of its own (P < 0.001).Table 2 compares the parameters found. The calculated MICs ofnitrous acid and the nitrite ion were 1.43 mg liter^–1and 2,033 mg liter^–1, respectively.

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TABLE 2. Comparison of the gamma model (reciprocal transformation) for action of Na nitrite and pH on growth of A. hydrophila

Nitrite, pH, and salt.
A series of checkerboard combinations of salt and NaNO₂ at aconstant pH were carried out. Model 7 was used to fit the data;parameters are given in Table 3.

Formula 7 (7)
The calculated MIC of NaNO₂ in combination withsalt decreases with pH, whereas that calculated for salt isindependent of pH for pHs of >5.5. At pH 5.5 the calculatedMIC for salt increased, but the confidence interval was alsovery large. Figure 4 compares the calculated MIC for total NaNO₂with the pHs from the experiments of nitrite/pH only and nitrite/saltand pH. The plot suggests that there are no changes in the calculatedMIC of total NaNO₂ in the presence of salt.

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TABLE 3. Iso-pH parameters for combinations of Na nitrite and salt^c

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FIG. 4. Comparison of the calculated MICs at various pHs for Na nitrite, analyzed alone ( ${circ}$ ) or in combination with salt (•). The bars give the upper and lower 95% confidence intervals.

The gamma model was fitted to the complete data set; the parametersobtained (Table 4) are in line with those expected for nitrousacid and nitrite. The parameters for salt are similar to thosefound previously (16).

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TABLE 4. Parameters for the gamma model for combinations of pH, Na nitrite, and salt^a

Surface response modeling.
Surface response modeling uses empirical models (polynomials)with an arbitrary set of constants, often chosen retrospectivelyto obtain a more "parsimonious" fit. The parameters obtainedfrom such a response surface fit are given in Table 5. The responsesurface used pH instead of the hydrogen ion concentration, sincethis improved the fit; the nitrite ion was not used, since thisgave rise to an ill-conditioned equation. All of the cross-termsbetween factors are significant (P < 0.01).

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TABLE 5. Response surface parameters for estimation of ln(TTD) values for combinations of pH, salt, and nitrous acid against A. hydrophila^a

Nominal logistic modeling.
The TTD data were degraded to nominal data: growth (G) = value1; no-growth (NG) = value 0. This allowed data which were previouslycensored to have some use. The logistic model used a simplefactorial model of pH, log of total nitrite, and log of salt.The prediction formula is given in Table 6. The contingencytable (Table 7) gives the discrepancies between the observedand calculated values. The model disagrees with the observedfor only six observations (1.2%). In all cases where growthwas observed but was modeled as NG, the TTD were close to theexperimental limit, and for those cases where no growth wasobserved but modeling showed G, the conditions were all "borderline,"i.e., in no case was a mismatch considered extraordinary.

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TABLE 6. Nominal logistic model for growth/no growth of A. hydrophila for combinations of pH, salt, and Na nitrite^a

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TABLE 7. Contingency table for observed G/NG and calculated G/NG^a

Comparison with the model of Palumbo et al.
A widely used predictive model for the growth of A. hydrophilais that of Palumbo et al. (29) (herein called the Palumbo model),which forms the basis of the predictive model published by theU.S. Department of Agriculture (Pathogen Modeling Program, version7.0; Eastern Regional Research Center, U.S. Department of Agriculture,Wyndmoor, PA) for this organism (although the model was updated[30]). The combined effects of temperature (5 to 42°C),NaCl (0.5 to 4.5%), pH (5.3 to 7.3), and NaNO₂ (0 to 200 µg/ml)on the aerobic growth of A. hydrophila K144 were studied. TheGompertz parameters B and M were modeled as either a quadraticor cubic polynomial function. Using the published quadraticmodel, the time to reach 7 log CFU ml^–1 for the rangeof temperature, pH, salt, and nitrite conditions given was foundusing the appropriate rearrangement of the Gompertz equation.

The hypothesis underlying the work given herein is that thegamma hypothesis is a valid framework for the investigationof the growth of microorganisms (36). In predictive models,which are based on the gamma approach, the temperature is modeledas a discrete function using, for example, the Rosso cardinaltemperature model (34). Using the published temperature (temp)cardinal parameters for A. hydrophila, the following model wasconstructed:

Formula 8 (8)
wheref(temp) is given by

Formula 8A (8a)
Themodel was compared to that of Palumbo et al. (29) in two ways:first a direct comparison with equation 8 using the parametersgiven in Table 4 and the published cardinal parameters (34)and second using these as the initial values for a nonlinearfitting of equation 8 to the TTD data calculated using the Palumbomodel. The parameters for the antimicrobial effect of the nitriteion were not used, since the maximum concentration of totalnitrite used by Palumbo et al. was 200 mg liter^–1.

In the first case, a reasonable fit to observations where thetemperature (T) was >5°C was found (r² = 0.79); the lackof fit for values where T equaled 5°C was not surprisinggiven that the model used the cardinal minimum T (T_min) valueof 5.1. By allowing the T_min parameter to relax, a good fitto the published data was obtained with an estimated T_min valueof –2.2°C. Allowing all of the parameters to relaxgave a better fit (Table 8).

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TABLE 8. Fit of the gamma model (log transform) to data from work of Palumbo et al. (29)

One thousand random values of temperature (range, 5 to 42),pH (5.5 to 6.8), NaCl (0.5 to 4.5%), and NaNO₂ (1 to 200 mgliter^–1) were produced and the ln(TTD) to 7 log CFU ml^–1calculated [ln(TTD₇)] using the Palumbo model. The logarithmicform of the gamma model (equation 1) was fitted to these calculateddata. Figure 5 shows a plot of the ln(TTD₇) calculated usingthe log model against that calculated from the Palumbo model(y = 0.955x + 0.191; r² = 0.96).

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FIG. 5. Comparison of the polynomial model by Palumbo et al. (29) with the gamma model (equation 1) for the log of the TTD of A. hydrophila in various environments of temperature, pH, salt, and nitrite concentration.

DISCUSSION

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Discussion

The analysis of the OD/time curves by using the integral functionor by simply calculating the area under the curve is usefulbecause it simplifies the presentation of data from multiplewells. In general, we have found that there are two major typesof OD/time plots—those which have congruent plots displaceddown the time axis with increasing inhibitor concentrations(which we have termed type 1 inhibitors) and those which havea similar initial TTD but have a decreasing maximum OD withincreasing inhibitor concentrations (termed type 2 inhibitors).If a single OD/time curve is assumed to follow a logistic-typecurve, then the integral of the logistic reveals that the gradientis related to the maximum OD attained.

When the pHs of solutions containing the same concentrationsof NaNO₂ were altered, the OD/time plots changed from type 2to type 1 as the pHs were reduced. This suggested that therewere two mechanisms of inhibition operating, dependent on pH.Since at a given pH, NaNO₂ exists in an equilibrium form betweenthe anion and nitrous acid, it was hypothesized that at thehigher pH the nitrite ion was the dominant inhibitor, whereasat the lower pH nitrous acid was dominant.

At a specific pH, no distinction between the actions of nitrousacid and nitrite is possible using a model which separates thetwo (e.g., equation 5), since they exist in equilibrium. Modelingsimply results in an exact correlation between the parametersfor the nitrite ion and those of nitrous acid. An assumptionmade in the modeling was that the combined effect (equation6) was applicable. Lambert and Lambert (17) have shown thatthis assumption is valid only under certain circumstances, e.g.,when the exponent powers are close to 1, or else a binomialtype of expansion is required. In this case, the assumptionled to a good fit of the model to the data. As expected, theMIC of total added nitrite decreases with decreasing pH. Itis noteworthy that this observation is missing from response-surfacemodels, and this reinforces the fact that such models are empiricisms.

Using data from the entire pH range, a distinction between theeffects of nitrous acid and the nitrite ion can be made. Thecalculated MIC (mg liter^–1) of nitrite indicates it isabout 1,400-fold less powerful than nitrous acid (2,033 versus1.43 mg liter^–1, respectively). From the data of Palumboet al. (29), the maximum concentration of NaNO₂ used was 200mg liter^–1, and so the nitrite ion would have had no observableaffect on inhibition. Also, the relevance of the nitrite ionper se as an antimicrobial is moot, since permitted levels oftotal NaNO₂ in foods are generally limited to less than 200mg kg^–1 (27).

Of relevance to the discussion, however, is whether the combinationof pH and NaNO₂ (as the total added) can be considered synergistic.The gamma model suggests that the antimicrobial effects of pHand the nitrite ion are independent, whereas the concentrationof nitrous acid (and remaining nitrite) is dependent on thepH. We would suggest that the label of antimicrobial synergybe retained only when the effect on the microbe is enhancedabove that expected due to the presence of both inhibitors.In this case, the effect of pH on nitrite is to produce morenitrous acid at a lower pH, which is simple physical chemistry,and therefore a label of antimicrobial synergy between pH andthe weak acid is not justified, since the action occurs whetheror not a microbe is present.

Combinations of nitrite with salt at a lowered pH are consideredin the literature to be synergistic against several organisms.Previous work in our laboratory showed that salt and pH haveindependent effects against A. hydrophila (16). In the workreported herein, the calculated MIC of total nitrite ion wasunaffected by the presence of salt. The gamma model fitted theobservations well, suggesting that combinations of pH, salt,and NaNO₂ have independent antimicrobial effects and that thegamma hypothesis is valid for these combinations.

Analyses of the data using a response-surface model or a nominallogistic model (for the growth/no-growth boundary) gave verygood fits to the data. In these cases significant cross-products(between factors) were recorded. This would appear to suggesta different conclusion as to the existence of interactions betweenfactors. The logarithmic transform of the gamma model with thepH, salt, and NaNO₂ data gave a root mean square error (RMSE)of 0.141 (nine parameters), whereas the response model had alower RMSE of 0.134 (10 parameters [Table 5]). We would arguethat the response surface fit, although a superior fit, is aninferior model to the gamma model (equation 1) because of anobjection given by Ratkowsky (32): that the parameters are notbiologically meaningful. A similar argument can be given tothe fit of the nominal logistic model: the fit is excellent(1.2% error) and there are numerous cross-terms, but the modelparameters are biologically meaningless.

We would disagree with Ratkowsky (32), however, and suggestthat polynomials do work—it simply depends on what youwant them to do. For the models described, a simple hierarchyconstruct is useful: the simplest is nominal logistic, followedby response surface, followed by mechanistic or truly predictivemodels. If the purpose of the model is to describe under whichconditions a likely product will fail, then a nominal logisticapproach is very useful (31). It gives the simple criterionof growth or no growth. The drawback is that the data need tocover the exact time (shelf-life) needed for the product. Thereis no information on when something will grow, since the datawill be recorded as G or NG. A recent example of this type ofmodeling used published data to define the boundaries of growthfor several pathogens (23).

If the time of growth of a particular combination of factorsis required, then a response surface model will be well suited.It can be interpolated, with the proviso as given by Baranyiet al. (2), and an error estimate can be given. The method hasthe drawback that the whole range of experiments have to becarried out first using a cogent experimental design, that extrapolationsare mathematically forbidden, and that addition of a new factormay require the entire design to be redone. A principal reasonfor the latter statement is the presence within the model ofinteraction terms.

A mechanistic model may have the advantage of a scientific basis,which the other types of models do not have. Furthermore, thistype of modeling can explore "inhibitory" space using the gammahypothesis as a directional tool. If the model is robust enough,it should be capable of extrapolation and expansion. With thisreasoning, the gamma model was supplemented with a cardinaltemperature function (34) with the assumption that the gammahypothesis was valid over the entire experimental temperaturerange, i.e., that temperature, pH, NaNO₂, and salt all act independentlyon the growth of A. hydrophila. The conclusion reached was thatthe fit of the gamma model (equation 1) to the data previouslyfound for pH, salt, and the total nitrite concentration, whenaugmented by a temperature function, can reproduce the datacalculated from the model of Palumbo et al. (29).

The Palumbo model contains several cross-terms, which were consideredto show interactive effects. That a gamma model fits the dataequally well suggests that in this case, there are no interactiveeffects.

Modeling microbiological data using the gamma approach can leadto a large savings in the time and resources required to producepredictive models for use in product development. If the gammahypothesis is valid, then careful experiments can be carriedout on individual factors; these can then be combined to givea larger model. This approach is wholly consistent with thehurdle approach, except that the gamma hypothesis negates thepossibility of synergistic effects. There may indeed be antimicrobialsynergistic effects, but within food microbiology there is little,if any, unequivocal evidence of such an occurrence. Part ofthe problem lies with how we define synergy and how we modelfor the effect, a problem that afflicts many other disciplinesdealing with antimicrobials (e.g., see references 11, 15, and28).

FOOTNOTES

* Corresponding author. Mailing address: Quality & Safety Department, Nestlé Research Centre, Vers-Chez-Les-Blanc, 1000 Lausanne 26, Switzerland. Phone: 41-21-785-8142. E-mail: ronald.lambert{at}rdls.nestle.com.

Published ahead of print on 9 February 2007.

REFERENCES

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