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Stochastic Model of Seed and Spore Germination

Experimentally determined germination curves of microbial spores come in two shapes, sigmoid and nonsigmoid, which can frequently be described by the stretched exponential model. The curve can also be simulated by a stochastic model where the germination probability can vary with time. This Demonstration shows that if the underlying probability of germination is constant, the germination curve is nonsigmoid. If the underlying germination probability rate function curve is itself sigmoid, so is the germination curve. If the germination probability rate function has a maximum, the germination curve can still be sigmoid but with an asymptotic germination level that depends on the peak's location, sharpness, and height.

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Snapshot 1: sigmoid germination curve with high germination level
Snapshot 2: constant germination curve with high germination level
Snapshot 3: peaked germination curve with low germination level
Germination curves of bacterial and fungal spores are frequently sigmoid in shape and can be described by the stretched exponential (Weibullian) model , where is the percent of germination at time , is the asymptotic fraction of germinated spores, is a characteristic time marking the inflection point, and , , is a "shape factor" representing the curve's steepness around the inflection point. The same model, but with , can be used to describe nonsigmoid curves, as in the germination of bacilli spores induced by ultra-high hydrostatic pressure.
Consider a spore dormant at time , . After time , it has a germination probability of , where is the probability rate function at time 1, and a probability to remain dormant; see the diagram above the plots. After a second , which for simplicity will remain the same, the germination probability is . Similarly, for the third time increment the probability will be , and so on. If, for instance, we assign a pertinent time unit, the germination probability after the time step will be . To create a discrete germination curve of a group of dormant spores, we start by generating a random number , , and check whether . If so, the spore is considered germinated and the calculation stops there. If not, then we generate another and check if , in which case the spore is considered germinated and the process ends for that particular spore. If not, the process is repeated with newly generated and until the spore germinates. The same is repeated for all spores and the tally is recorded. The discrete germination curve is the sum of the number of germinated spores at 0, 1, 2, … time units. The data so produced can then be fitted with the phenomenological model using the Mathematica built-in function NonlinearModelFit and the resulting curve superimposed on the data.
This Demonstration allows you to choose the type of , sigmoid, constant, or peaked, with a setter bar. Using sliders you can select the seed for the random number generation, the number of points to generate, the initial number of dormant spores, and the parameters of the probability rate function . The program then plots the probability rate function and below it the corresponding discrete and fitted germination curve, and displays the percent germination at the chosen time. You can vary the plots' axes maxima with sliders.
The objective of this Demonstration is not to match any particular germination pattern but only to provide visualization of the relationship between the stochastic and phenomenological models. As the number of spores increases the curve created by the stochastic model becomes smoother and more deterministic. Although the Demonstration addresses the germination of bacterial and fungal spores, it may be just as relevant to the germination of plant seeds.
Reference
[1] M. Peleg and M. D. Normand, "Modeling of Fungal and Bacterial Spore Germination under Static and Dynamic Conditions," Applied and Environmental Microbiology, 79(21), 2013 pp. 6765–6775. doi:10.1128/AEM.02521-13.
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