Modeling of fungal and bacterial spore germination under static and dynamic conditions

Abstract

Isothermal germination curves, sigmoid and nonsigmoid, can be described by a variety of models reminiscent of growth models. Two of these, which are consistent with the percent of germinated spores being initially zero, were selected: one, Weibullian (or "stretched exponential"), for more or less symmetric curves, and the other, introduced by Dantigny's group, for asymmetric curves (P. Dantigny, S. P.-M. Nanguy, D. Judet-Correia, and M. Bensoussan, Int. J. Food Microbiol. 146:176-181, 2011). These static models were converted into differential rate models to simulate dynamic germination patterns, which passed a test for consistency. In principle, these and similar models, if validated experimentally, could be used to predict dynamic germination from isothermal data. The procedures to generate both isothermal and dynamic germination curves have been automated and posted as freeware on the Internet in the form of interactive Wolfram demonstrations. A fully stochastic model of individual and small groups of spores, developed in parallel, shows that when the germination probability is constant from the start, the germination curve is nonsigmoid. It becomes sigmoid if the probability monotonically rises from zero. If the probability rate function rises and then falls, the germination reaches an asymptotic level determined by the peak's location and height. As the number of individual spores rises, the germination curve of their assemblies becomes smoother. It also becomes more deterministic and can be described by the empirical phenomenological models.

Fig 1

Published sigmoid and nonsigmoid isothermal germination data (solid circles) fitted with equations 4 and 5 as models shown as solid gray and black lines, respectively. (Top) Sigmoid curve of fungal spores (m > 1). (Bottom) Nonsigmoid curve for bacillus spores exposed to high hydrostatic pressure fitted with equation 4 with m = 1. The regression coefficients, r², are all in the range of 0.991 to 0.999. The fungi's data are from the work of Gougouli and Koutsoumanis (10), and the bacilli's are from the work of Wei et al. (13).

Fig 2

Screen display of the Wolfram demonstration “Isothermal Germination of Seeds and Microbial Spores” set for models A (left) and B (right) at m > 1. Note that the model parameters can be entered and varied with sliders.

Fig 3

Screen display of the Wolfram demonstration “Isothermal Germination of Seeds and Microbial Spores” set for models A (left) and B (right) at m < 1. Note that the model parameters can be entered and varied with sliders.

Fig 4

Hypothetical temperature dependencies of the germination parameters P_asym, t_c, and m in equations 4 and 5.

Fig 5

Comparison of generated dynamic germination curves for a quasi-isothermal temperature profile and corresponding truly isothermal curve, which has been superimposed. The dashed curves were generated for T(t) = 20.0 using equations 6 and 7 as models. The superimposed solid gray curves were generated for the quasi-isothermal profile T(t) = 20.0 + 10⁻⁸sin(0.2t) as the numerical solution of the differential rate model, equation 8 or 9. Note that the curves generated with the two models are indistinguishable.

Fig 6

Probabilities tree (Markov chain) of a single germinating dormant spore.

Fig 7

Simulated hypothetical oscillating temperature profiles (top) and corresponding germination curves (middle) and germination rate curves (bottom), generated with equations 8 (left) and 9 (right) as models with m > 1.

Fig 8

Simulated hypothetical oscillating temperature profiles (top) and corresponding germination curves (middle) and germination rate curves (bottom), generated with equations 8 (left) and 9 (right) as models, with m < 1.

Fig 9

Screen display of the Wolfram demonstration “Non-Isothermal Germination of Seeds and Microbial Spores” set for model A with m > 1. Note that the model parameters can be entered and varied with sliders.

Fig 10

Examples of raw germination curves of 10, 30, 100, and 500 dormant spores, generated with the stochastic model (equation 12). Notice that as the number of spores in the population rises, the germination curve becomes smoother and more deterministic.

Fig 11

Germination probability rate function effect on the germination curve's shape. The filled circles are data generated with the stochastic model, and the superimposed solid gray curve is the fit of model A (equation 4). Notice that a constant (or falling) probability rate produces a nonsigmoid germination curve.

Fig 12

Germination probabilities tree when cell division and mortality occur simultaneously.