Evaluation of a stochastic inactivation model for heat-activated spores of Bacillus spp

Abstract

Heat activates the dormant spores of certain Bacillus spp., which is reflected in the "activation shoulder" in their survival curves. At the same time, heat also inactivates the already active and just activated spores, as well as those still dormant. A stochastic model based on progressively changing probabilities of activation and inactivation can describe this phenomenon. The model is presented in a fully probabilistic discrete form for individual and small groups of spores and as a semicontinuous deterministic model for large spore populations. The same underlying algorithm applies to both isothermal and dynamic heat treatments. Its construction does not require the assumption of the activation and inactivation kinetics or knowledge of their biophysical and biochemical mechanisms. A simplified version of the semicontinuous model was used to simulate survival curves with the activation shoulder that are reminiscent of experimental curves reported in the literature. The model is not intended to replace current models to predict dynamic inactivation but only to offer a conceptual alternative to their interpretation. Nevertheless, by linking the survival curve's shape to probabilities of events at the individual spore level, the model explains, and can be used to simulate, the irregular activation and survival patterns of individual and small groups of spores, which might be involved in food poisoning and spoilage.

FIG. 1.

A schematic view of a survival curve having an activation shoulder. S(t) is the ratio between the number N(t) of viable spores at time t and the initial number N₀. Notice the discrepancy between the two ways to estimate the number of dormant spores, represented by the dashed and dotted gray lines.

FIG. 2.

Demonstration of the fit of equation 1 (solid line) and equation 2 (dashed line) to survival curves of B. stearothermophilus spores at two temperatures. Notice the postpeak concavity of the curves. In such cases, the estimated number of dormant spores reached by the tangent method will depend on the experiment duration. The original experimental data are from Sapru et al. (25).

FIG. 3.

Schematic view of the inactivation of an active (or activated) spore (top) and the inactivation/activation of a dormant spore (bottom).

FIG. 4.

Probability tree of an active or activated spore.

FIG. 5.

Two Weibullian curves and one log-linear survival curve of active or activated spores generated with equation 4 as a model. Notice that a monotonically increasing mortality probability rate P_m(t) (top) produces a survival curve with downward concavity when plotted on semilogarithmic coordinates (bottom). Decreasing mortality probability rate produces a survival curve with upward concavity. A log-linear survival curve is a special case where the mortality probability rate is constant. Also notice that the difference between the three modes of inactivation is hardly revealed when the survival curves are plotted on linear coordinates (middle).

FIG. 6.

Simulated survival curves of groups of 5 and 50 active spores generated with the discrete version of the survival model (equation 4). The two curves marked with Roman numerals are different runs representing replicates. Notice that as the number of spores increases, the curve becomes smoother, and the survival pattern is more deterministic.

FIG. 7.

Probability tree of a dormant spore.

FIG. 8.

Simulated survival and activation curves of groups of 5, 10, and 50 dormant spores generated with the discrete version of the activation/survival model (equation 6). The three curves marked with Roman numerals are different runs representing replicates. Notice that as the number of spores increases, the curves become smoother, and the patterns are more deterministic.

FIG. 9.

Simulated activation/inactivation curves of mixed groups of active and dormant spores (ratio of 1 to 5) generated with the two discrete versions of the model (equations 4 and 6). The three curves marked with Roman numerals are different runs representing replicates. Notice that as the number of spores increases, the curves become smoother, and the activation/survival pattern is more deterministic.

FIG. 10.

Simulated activation/inactivation curves of 100 dormant spores generated with the semicontinuous version of the model (equation 7) plotted on linear and semilogarithmic coordinates. Notice that a very similar survival pattern would be observed had a fraction of initially active spores been added.

FIG. 11.

Activation/survival of B. stearothermophilus spores at 105 and 110°C described by the semicontinuous version of equation 7 (top) and the corresponding activation and mortality rate functions P_a(t) and P_m(t), respectively (bottom). The original data are from Sapru et al. (24).