Maria G. Corradini

Generating microbial survival curves during thermal processing in real time

Aims:  To develop a method to calculate and record theoretical microbial survival curves during thermal processing of foods and pharmaceutical products simultaneously with the changing temperature. Moreover, to demonstrate that the method can be used to calculate nonisothermal survival curves, with widely available software such as Microsoft Excel®.

The Arrhenius Equation Revisited

The Arrhenius equation has been widely used as a model of the temperature effect on the rate of chemical reactions and biological processes in foods. Since the model requires that the rate increase monotonically with temperature, its applicability to enzymatic reactions and microbial growth, which have optimal temperature, is obviously limited. This is also true for microbial inactivation and chemical reactions that only start at an elevated temperature, and for complex processes and reactions that do not follow fixed order kinetics, that is, where the isothermal rate constant, however defined, is a function of both temperature and time. The linearity of the Arrhenius plot, that is, Ln[k(T)] vs. 1/T where T is in °K has been traditionally considered evidence of the models validity. Consequently, the slope of the plot has been used to calculate the reaction or processes’ “energy of activation,” usually without independent verification. Many experimental and simulated rate constant vs. temperature relationships that yield linear Arrhenius plots can also be described by the simpler exponential model Ln[k(T)/k(Treference)] = c(T-Treference). The use of the exponential model or similar empirical alternative would eliminate the confusing temperature axis inversion, the unnecessary compression of the temperature scale, and the need for kinetic assumptions that are hard to affirm in food systems. It would also eliminate the reference to the Universal gas constant in systems where a “mole” cannot be clearly identified. Unless proven otherwise by independent experiments, one cannot dismiss the notion that the apparent linearity of the Arrhenius plot in many food systems is due to a mathematical property of the models equation rather than to the existence of a temperature independent “energy of activation.” If T+273.16°C in the Arrhenius models equation is replaced by T+b, where the numerical value of the arbitrary constant b is substantially larger than T and Treference, the plot of Ln k(T) vs. 1/(T+b) will always appear almost perfectly linear. Both the modified Arrhenius model version having the arbitrary constant b, Ln[k(T)/k(Treference) = a[1/ (Treference+b)-1/ (T+b)], and the exponential model can faithfully describe temperature dependencies traditionally described by the Arrhenius equation without the assumption of a temperature independent “energy of activation.” This is demonstrated mathematically and with computer simulations, and with reprocessed classical kinetic data and published food results.

Estimating non-isothermal bacterial growth in foods from isothermal experimental data

Aim:  To develop a mathematical method to estimate non‐isothermal microbial growth curves in foods from experiments performed under isothermal conditions and demonstrate the methods applicability with published growth data.

Comparing and correlating solubility parameters governing the self-assembly of molecular gels using 1,3:2,4-dibenzylidene sorbitol as the gelator.

Solvent properties play a central role in mediating the aggregation and self-assembly of molecular gelators and their growth into fibers. Numerous attempts have been made to correlate the solubility parameters of solvents and gelation abilities of molecular gelators, but a comprehensive comparison of the most important parameters has yet to appear. Here, the degree to which partition coefficients (log P), Henry’s law constants (HLC), dipole moments, static relative permittivities (εr), solvatochromic ET(30) parameters, Kamlet–Taft parameters (β, α, and π), Catalan’s solvatochromic parameters (SPP, SB, and SA), Hildebrand solubility parameters (δi), and Hansen solubility parameters (δp, δd, δh) and the associated Hansen distance (Rij) of 62 solvents (covering a wide range of properties) can be correlated with the self-assembly and gelation of 1,3:2,4-dibenzylidene sorbitol (DBS) gelation, a classic molecular gelator, is assessed systematically. The approach presented describes the basis for each of the parameters and how it can be applied. As such, it is an instructional blueprint for how to assess the appropriate type of solvent parameter for use with other molecular gelators as well as with molecules forming other types of self-assembled materials. The results also reveal several important insights into the factors favoring the gelation of solvents by DBS. The ability of a solvent to accept or donate a hydrogen bond is much more important than solvent polarity in determining whether mixtures with DBS become solutions, clear gels, or opaque gels. Thermodynamically derived parameters could not be correlated to the physical properties of the molecular gels unless they were dissected into their individual HSPs. The DBS solvent phases tend to cluster in regions of Hansen space and are highly influenced by the hydrogen-bonding HSP, δh. It is also found that the fate of this molecular gelator, unlike that of polymers, is influenced not only by the magnitude of the distance between the HSPs for DBS and the HSPs of the solvent, Rij, but also by the directionality of Rij: if the solvent has a larger hydrogen-bonding HSP (indicating stronger H-bonding) than that of the DBS, then clear gels are formed; opaque gels form when the solvent has a lower δh than does DBS.

Microbial Growth Curves: What the Models Tell Us and What They Cannot

Most of the models of microbial growth in food are Empirical algebraic, of which the Gompertz model is the most notable, Rate equations, mostly variants of the Verhulsts logistic model, or Population Dynamics models, which can be deterministic and continuous or stochastic and discrete. The models of the first two kinds only address net growth and hence cannot account for cell mortality that can occur at any phase of the growth. Almost invariably, several alternative models of all three types can describe the same set of experimental growth data. This lack of uniqueness is by itself a reason to question any mechanistic interpretation of growth parameters obtained by curve fitting alone. As argued, all the variants of the Verhulsts model, including the Baranyi-Roberts model, are empirical phenomenological models in a rate equation form. None provides any mechanistic insight or has inherent advantage over the others. In principle, models of all three kinds can predict non-isothermal growth patterns from isothermal data. Thus a modeler should choose the simplest and most convenient model for this purpose. There is no reason to assume that the dependence of the “maximum specific growth rate” on temperature, pH, water activity, or other factors follows the original or modified versions of the Arrhenius model, as the success of Ratkowskys square root model testifies. Most sigmoid isothermal growth curves require three adjustable parameters for their mathematical description and growth curves showing a peak at least four. Although frequently observed, there is no theoretical reason that these growth parameters should always rise and fall in unison in response to changes in external conditions. Thus quantifying the effect of an environmental factor on microbial growth require that all the growth parameters are addressed, not just the “maximum specific growth rate.” Different methods to determine the “lag time” often yield different values, demonstrating that it is a poorly defined growth parameter. The combined effect of several factors, such as temperature and pH or aw, need not be “multiplicative” and therefore ought to be revealed experimentally. This might not be always feasible, but keeping the notion in mind will eliminate theoretical assumptions that are hard to confirm. Modern mathematical software allows to model growing or dying microbial populations where cell division and mortality occur simultaneously and can be used to explain how different growth patterns emerge. But at least in the near future, practical problems, like translating a varying temperature into a corresponding microbial growth curve, will be solved with empirical rate models, which despite not being “mechanistic” are perfectly suitable for this purpose.

Demonstration of the applicability of the Weibull-log-logistic survival model to the isothermal and nonisothermal inactivation of Escherichia coli K-12 MG1655.

Published isothermal semilogarithmic survival curves of Escherichia coli K-12 MG1655, in the range of 49.8 to 60.6 degrees C, all had noticeable downward concavity. They could be described by the model log S(t) = -b(T)t n, where S(t) = N(t)/N0, N(t) and N0 being the momentary and initial number of organisms, respectively; b(T) is a temperature-dependent rate parameter; and n is a constant found to be about 1.5. The temperature dependence of b(T) could be described by the log-logistic model, b(T) = ln[1 + exp[k(T - Tc)]], which had an almost perfect fit, with k = 0.88 degrees C(-1) and Tc = 60.5 degrees C. The constants, n, k, and Tc were considered the organisms survival parameters in the particular medium. They were incorporated into a rate equation on the assumption that in nonisothermal heating, the momentary inactivation rate is the isothermal rate at the momentary temperature at a time that corresponds to the momentary survival ratio. This models estimates matched the actual survival curves obtained in the same work under two different nonisothermal heating profiles, lending support to the notion that the Weibull-log-logistic model combination can be used not only to describe isothermal inactivation mathematically, but also to predict survival patterns under nonisothermal conditions.

Linear and Non-Linear Kinetics in the Synthesis and Degradation of Acrylamide in Foods and Model Systems

Isothermal acrylamide formation in foods and asparagine-glucose model systems has ubiquitous features. On a time scale of about 60 min, at temperatures in the approximate range of 120−160°C, the acrylamide concentration-time curve has a characteristic sigmoid shape whose asymptotic level and steepness increases with temperature while the time that corresponds to the inflection point decreases. In the approximate range of 160−200°C, the curve has a clear peak, whose onset, height, width and degree of asymmetry depend on the systems composition and temperature. The synthesis-degradation of acrylamide in model systems has been recently described by traditional kinetic models. They account for the intermediate stages of the process and the fate of reactants involved at different levels of scrutiny. The resulting models have 2–6 rate constants, accounting for both the generation and elimination of the acrylamide. Their temperature dependence has been assumed to obey the Arrhenius equation, i.e., each step in the reaction was considered as having a fixed energy of activation. A proposed alternative is constructing the concentration curve by superimposing a Fermian decay term on a logistic growth function. The resulting model, which is not unique, has five parameters: a hypothetical uninterrupted generation-level, two steepness parameters; of the concentration climbs and fall and two time characteristics; of the acrylamide synthesis and elimination. According to this model, peak concentration is observed only when the two time constants are comparable. The peaks shape and height are determined by the gap between the two time constants and the relative magnitudes of the two “rate” parameters. The concept can be extended to create models of non-isothermal acrylamide formation. The basic assumption, which is yet to be verified experimentally, is that the momentary rate of the acrylamide synthesis or degradation is the isothermal rate at the momentary temperature, at a time that corresponds to its momentary concentration. The theoretical capabilities of a model of this kind are demonstrated with computer simulations. If the described model is correct, then by controlling temperature history, it is possible to reduce the acrylamide while still accomplishing much of the desirable effects of a heat process.

Dynamic Model of Heat Inactivation Kinetics for Bacterial Adaptation

ABSTRACT The Weibullian-log logistic (WeLL) inactivation model was modified to account for heat adaptation by introducing a logistic adaptation factor, which rendered its “rate parameter” a function of both temperature and heating rate. The resulting model is consistent with the observation that adaptation is primarily noticeable in slow heat processes in which the cells are exposed to sublethal temperatures for a sufficiently long time. Dynamic survival patterns generated with the proposed model were in general agreement with those of Escherichia coli and Listeria monocytogenes as reported in the literature. Although the modified models rate equation has a cumbersome appearance, especially for thermal processes having a variable heating rate, it can be solved numerically with commercial mathematical software. The dynamic model has five survival/adaptation parameters whose determination will require a large experimental database. However, with assumed or estimated parameter values, the model can simulate survival patterns of adapting pathogens in cooked foods that can be used in risk assessment and the establishment of safe preparation conditions.

Estimating the Heat Resistance Parameters of Bacterial Spores from their Survival Ratios at the End of UHT and other Heat Treatments

Accurate determination of bacterial cells or the isothermal survival curves of spores at Ultra High Temperatures (UHT) is hindered by the difficulty in withdrawing samples during the short process and the significant role that the come up and cooling times might play. The problem would be avoided if the survival parameters could be derived directly from the final survival ratios of the non-isothermal treatments but with known temperature profiles. Non-linear inactivation can be described by models that have at least three survival parameters. In the simplified version of the Weibullian –log logistic model they are n, representing the curvature of the isothermal semilogarithmic survival curves, Tc, a marker of the temperature where the inactivation accelerates and k, the slope of the rate parameter at temperatures well above Tc. In principle, these three unknown parameters can be calculated by solving, simultaneously, three rate equations constructed for three different temperature profiles that have produced three corresponding final survival ratios, which are determined experimentally. Since the three equations are constructed from the numerical solutions of three differential equations, this might not always be a practical option. However, the solution would be greatly facilitated if the problem could be reduced to the solution of only two simultaneous equations. This can be done by progressively changing the value of n by small increments or decrements and solving for k and Tc. The iterations continue until the model constructed with the calculated k and Tc values correctly predicts the survival ratio obtained in a third heat treatment with a known temperature profile. Once n, k, and Tc are established in this way, the resulting model can be used to predict the complete survival curves of the organism or spore under almost any contemplated or actual UHT treatment in the same medium. The potential of the method is demonstrated with simulated inactivation patterns and its predictive ability with experimental survival data of Bacillus sporothermodurans. Theoretically at least, the shown calculation procedure can be applied to other thermal preservation methods and to the prediction of collateral biochemical reactions, like vitamin degradation or the synthesis of compounds that cause discoloration. The concept itself can also be extended to non-Weibullian inactivation or synthesis patterns, provided that they are controlled by only three or fewer kinetic parameters.

Probabilistic Model of Microbial Cell Growth, Division, and Mortality

ABSTRACT After a short time interval of length δt during microbial growth, an individual cell can be found to be divided with probability Pd(t)δt, dead with probability Pm(t)δt, or alive but undivided with the probability 1 − [Pd(t) + Pm(t)]δt, where t is time, Pd(t) expresses the probability of division for an individual cell per unit of time, and Pm(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitats properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the populations size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a “shoulder,” and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With Pd(t) and Pm(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.