Modeling of Fungal and Bacterial Spore Germination under Static andDynamic Conditions

Micha Peleg, Mark D. Normand

Department of Food Science, University of Massachusetts, Amherst, Massachusetts, USA

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 asymmetriccurves (P. Dantigny, S. P.-M. Nanguy, D. Judet-Correia, and M. Bensoussan, Int. J. Food Microbiol. 146:176 –181, 2011). Thesestatic models were converted into differential rate models to simulate dynamic germination patterns, which passed a test forconsistency. In principle, these and similar models, if validated experimentally, could be used to predict dynamic germinationfrom isothermal data. The procedures to generate both isothermal and dynamic germination curves have been automated andposted as freeware on the Internet in the form of interactive Wolfram demonstrations. A fully stochastic model of individual andsmall groups of spores, developed in parallel, shows that when the germination probability is constant from the start, the germi-nation curve is nonsigmoid. It becomes sigmoid if the probability monotonically rises from zero. If the probability rate functionrises and then falls, the germination reaches an asymptotic level determined by the peak’s location and height. As the number ofindividual spores rises, the germination curve of their assemblies becomes smoother. It also becomes more deterministic andcan be described by the empirical phenomenological models.

The germination of Bacillus and Clostridium endospores canplay an important role in food safety and, as in the case of

anthrax, in bioterrorism and public health as well. But germina-tion can also influence health promotion, as in the case of supple-menting food with probiotic bacilli (1). It is no wonder, therefore,that the biochemical and biophysical mechanisms of bacterialspore germination and its kinetics have been intensively studied(2–6). Fungal spore germination can play an important role infood production (notably in blue cheeses) and spoilage but also inskin and other human diseases. Thus, the mechanisms of yeast andmold spore germination, and their kinetics, have also been exten-sively studied (7, 8). Although microbial cell division and sporegermination are distinct physiological phenomena, their timescales can be about the same. The two processes’ kinetics may alsohave superficial similarities. When the percentage (or fraction of)germinated microbial spores is plotted versus time, the typicalresult is a sigmoid curve, which resembles the isothermal growthcurve of microbial cells in a closed habitat prior to the onset ofmortality. It has therefore been tempting to use microbial growthmodels, notably the Gompertz model (see below), to describe iso-thermal germination curves and extend their application to dy-namic conditions such as fluctuating temperatures.

The main objectives of this work were as follows: (i) to developa mathematical method to convert models describing isothermalgermination curves into rate models applicable to dynamic con-ditions; (ii) to automate the procedures of generating static anddynamic germination curves and make them available as freelydownloadable interactive software on the Internet; and (iii) todevelop a fully stochastic model to explain observed germinationpatterns (isothermal and dynamic) in terms of the time dependentgermination probabilities of individual spores.

Theoretical background. A static germination curve is a plotof the percent germinated bacterial, fungal, or yeast spores as afunction of time under constant temperature, humidity, pH, etc.(In botany or agriculture, a germination curve refers to the per-

cent of sprouted seeds as a function of time.) In this article, weconsider “static” synonymous with “isothermal,” assuming thatall other factors that affect the germination kinetics remain un-changed. Since germination of an individual spore (and a plantseed) is a process rather than an instantaneous event, the time atwhich a spore is considered germinated depends on the criterionused to define this state. Consequently, different criteria, such asones based on the spore’s size or stem’s length, for example, willresult in a shifted germination curve (8). Since the focus of thiswork is on the mathematical characterization of germinationcurves, we do not address this issue, and we consider the germi-nated state as defined by the authors of the cited works. Typically,a microbial germination curve has a sigmoid shape (9–11), al-though nonsigmoid shapes are sometimes also encountered (12,13) (Fig. 1; also, see below).

A typical sigmoid germination curve can be described by thepopular and flexible Gompertz growth model, which whenadapted to germination would have the form

LogP�t��Pasym � C exp��exp��B�t � M��� (1)

where Pasym is the asymptotic germination level (percent), M is theinflection point of the curve (having time units), B is the slope ofthe curve at M, and C is the difference between the asymptotic andinitial germination levels (percent). A modified form of the Gom-pertz model was adapted for germination by Gougouli and Kout-

Received 26 July 2013 Accepted 23 August 2013

Published ahead of print 30 August 2013

Address correspondence to Micha Peleg, micha.peleg@foodsci.umass.edu.

This paper is a contribution of the Massachusetts Agricultural ExperimentalStation at Amherst.

Copyright © 2013, American Society for Microbiology. All Rights Reserved.

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soumanis (10). It was adapted from the work of Zwietering et al.(14) and has the form

P�t� � Pasymexp��exp� �ge

Pasym��g � t� � 1 (2)

where �g is the germination rate and the inflection point at t � �g.Both versions of the Gompertz model have a problem. It is

neither the cumbersome formula nor its fit to experimental iso-thermal data, which is almost always excellent as judged by statis-tical criteria. The problem is the self-contradiction that at t � 0,P(0) � 0 [or, in growth curve descriptions, that the initial numberof cells N(0) � N0]. This has little influence on the model’s degreeof fit to static data, but it becomes a serious drawback when onetries to establish a boundary condition in trying to develop a dy-namic rate model (15) (see below). The same can be said aboutvarious versions of the general logistic (Richard’s) model, whichfor initially ungerminated spores might assume the form

P�t� �Pasym

1 � exp�B�M � t�� (3)

using the same definitions of B and M as in equation 1.To avoid the problem, we have chosen two germination mod-

els. The first, model A, is the cumulative form of the Weibulldistribution function (or “stretched exponential”), which for ourpurposes can be written as

P�t� � Pasym�1 � exp���t ⁄ tc�m�� (4)

where Pasym (0 � Pasym � 100) is the asymptotic percent germi-nation, tc is a characteristic time or “location factor,” and m is adimensionless “shape factor,” which primarily controls the steep-ness, or span, of the germination curve P(t). By definition, at t � 0,P(0) � 0, at tc, P(tc) � Pasym (1 � 1/e), and as t approaches infinity,P(t) approaches Pasym. Notice that the inflection point accordingto this model is not at tc but at t � tc ((m � 1)/m)1/m.

The other model, which we call model B, was recently intro-duced to germination kinetics by Dantigny et al. (16) and has theform

P�t� � Pasym�1 � 1 ⁄ �1 � �t ⁄ tc�m�� (5)

where Pasym, tc, and m have the same assignments as in equation 4.Here, too, P(0) � 0 by definition, and as t approaches infinity, P(t)approaches Pasym. However, at tc, P(tc) � Pasym/2, and the inflec-tion point is at t � tc ((m � 1)/(m � 1))1/m.

For t � 0, both models produce a sigmoid curve whenever m is

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 withequation 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).

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�1 and nonsigmoid when 0 � m � 1. The main difference be-tween model A and model B is that with the same germinationparameters Pasym, tc, and m, the germination curves generated bythe latter are more skewed to the right than those generated by theformer (see below).

The interested reader can generate, inspect, and compare othergermination curves produced by these two models having differentparameter combinations with the freely downloadable Wolframdemonstration entitled “Isothermal Germination of Seeds and Mi-

crobial Spores,” available at http://demonstrations.wolfram.com/IsothermalGerminationOfSeedsAndMicrobialSpores/.

Screen displays of the demonstrations for m values of �1 and�1 are shown in Fig. 2 and 3. The demonstration is part of theWolfram Demonstrations Project sponsored by Wolfram Re-search (Champaign, IL), the author of Mathematica, the programused in this work. To date, the project has over 9,000 demonstra-tions in a variety of scientific and other fields. The Wolfram CDFPlayer, the program that runs the demonstrations, is freely down-

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. Notethat 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. Notethat the model parameters can be entered and varied with sliders.

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loadable following instructions on the screen. It can also be used torun and manipulate all the demonstrations in the project.

The fit of the two models to published isothermal germinationdata is demonstrated in Fig. 1. Although this section specificallyaddresses isothermal germination models, much of it is relevant tostatic environmental conditions in general, especially concerningmoisture content, pH, and the presence of chemical triggers.These, together with temperature, determine the spore’s germina-tion pattern, which is characterized by the kind of model (sigmoidor nonsigmoid, symmetric or skewed around the inflection point)and the magnitude of germination parameters, Pasym, tc, and m.With both models, the effects of inhibitory factors would be gen-erally manifested in lowering the magnitude of Pasym (to zero inthe case of total inhibition), increasing the magnitude of tc, andlowering m. In contrast, germination-promoting factors wouldhave the opposite effect on these parameters’ magnitudes.

The two germination models (equations 4 and 5) can be usedinterchangeably where the germination curve is steep, i.e., resem-bling a step function. The difference between them becomes sub-stantial where the germination curves’ symmetry or asymmetrybecomes noticeable, in which case the choice between them can bebased on their degree of fit to the experimental data at and asjudged by statistical criteria.

Germination as a stochastic process. A germination curvepresented as the relationship of percent germinated spores versustime can be considered the cumulative density function (CDF) ofthe germination events’ temporal distribution. Thus, not surpris-ingly, one can find reports where the germination progress is pre-sented as a histogram of the number of spores germinated at par-ticular times (17). In the case of a sigmoid germination curve,which is the CDF of the time-to-germination unimodal distribu-tion, such a histogram is expected to have a typical peaked shape.The probability density function (PDF) form of the distribution,as its name implies, also indicates that if the germination curve issigmoid, the probability that an individual spore will be foundgerminated is not constant but varies with time (see below).

MATERIALS AND METHODSDevelopment of a deterministic model of dynamic germination. Con-sider a population of initially dormant spores whose isothermal germina-tion curves in the pertinent temperature range follow equation 4 or equa-tion 5 as a model. The reference to a pertinent temperature range is tostress that neither the dormant spores nor the vegetative cells, which thosethat have been germinated produced, are inactivated during the process.We also assume for the sake of simplicity that any cell division that mightfollow the germination is not a factor which affects the germinated sporescount, a scenario to which we refer below.

In principle, all three germination parameters in the two models thatwe have chosen can be temperature dependent, in which case equations 4and 5 should be written as follows:

P�T� � Pasym�T��1 � exp��� t

tc�T�m�T�� (6)

and

P�T� � Pasym�T� 1 �1

1 � � t

tc(T)�m�T�� (7)

The temperature dependencies of the germination parameters Pasym(T),tc(T), and m(T) would depend on the particular spore type, history, andgermination medium. In general, though, at optimal or near-optimal condi-tions, they would have a form similar to that shown in Fig. 4. Note that not allmicrobial spores reach 100% germination even under optimal conditions, inwhich case the asymptote of Pasym(T) will be at a lower value.

As in modeling of dynamic microbial growth and inactivation (18), weassume that under nonisothermal conditions, where the temperature pro-file is T(t), the momentary germination rate at time t is the isothermalgermination rate at the momentary temperature T(t), at the time t*(t) thatcorresponds to the momentary germination level, P(t). Since the momen-tary parameters are Pasym(t) � Pasym(T(t)), tc(t) � tc (T(t)), and m(t) �m(T(t)), the momentary isothermal germination rate according to equa-tions 6 and 7 is

dP�t�dt

�

Pasym�t�exp��� t��t�tc�t� �

t�m�t�� t��t�tc�t� m�t� � 1

tc�t� (8)

and

dP�t�dt

�

m�t�Pasym�t�� t��t�tc�t�

�1�m�t�

�1 � � t��t�tc�t� m�t�2

tc�t�(9)

where t*(t), the inverse of equation 6 and 7, is

t��t� �� log�exp��� t

tc�t�m�t�� 1

m�t�tc�t� (10)

and

t��t� � tc�t�� Pasym

Pasym � P�t� � 1 1m�t�

(11)

respectively.Despite their cumbersome appearance, equations 8 and 9 are both

ordinary differential equations (ODE). Hence, they can be rapidly solvednumerically by Mathematica, the program used in this work, and othercommercial mathematical programs, with the boundary condition beingthat at t � 0, P(0) � 0. The difference between these rate models and

FIG 4 Hypothetical temperature dependencies of the germination parameters Pasym, tc, and m in equations 4 and 5.

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germination models based on the isothermal equation (10, 11) is that withequations 8 to 11, one can generate germination curves for continuouslychanging temperature profiles rather than being limited to temperaturehistories consisting of constant temperature segments.

To test the procedure’s performance, we ran the program with tem-perature profiles such as T(t) � T0 � 10�8Sin[0.05t], which for all prac-tical purposes are the same as an isothermal process where T(t) � T0. Thisallowed us to compare curves produced by the numerical solution ofequations 8 and 9 with curves produced by the corresponding algebraicisothermal models, equations 6 and 7, for T � T0. As expected, the curvesgenerated by the two methods were indistinguishable regardless of themodel chosen (A or B) and whether the shape factor m was larger orsmaller than 1.0, as demonstrated in Fig. 5.

Development of a stochastic model. Consider an individual sporedormant at t � 0. As shown in Fig. 6 (left), after time t it has a probabilityPg(1)t1 of being found germinated and 1 � Pg(1)t1 of remaining dor-mant, where Pg(1) is the germination’s probability rate function. Noticethat while Pg(1) can assume values larger than 1, the product Pg(1)t1

must satisfy the condition 0 � Pg(1)t1 �1. The same applies to all sub-sequent germination probabilities. After a second time interval, t2

(Fig. 6, right), if the spore is still dormant, it would have the probability of(1 � Pg(1)t1)Pg(2)t2 of germinating. If still dormant, then after a thirdtime interval (t3), it would have the probability (1 � Pg(1) t1)(1 �Pg(2) t2)Pg(3)t3 of germinating, and so on for t4, etc., where Pg(1),

Pg(2), Pg(3), Pg(4). . . are the germination probability rate functions at thecorresponding times, and t1, t2, t3, t4. . . are the time intervals be-tween subsequent observations. For convenience, we assign to all the t’sa pertinent unit time, i.e., ti � 1. Thus, the probability of finding aninitially dormant spore germinated after time t (t � i 1) will be given bythe Markov chain, or probabilities tree, shown schematically in Fig. 6, thatis

P�0� � 0

P�1� � Pg�1�P�2� � �1 � Pg�1��Pg�2�P�3� � �1 � Pg�1�� �1 � Pg�2��Pg�3�P�4� � �1 � Pg�1�� �1 � Pg�2�� �1 � Pg�3��Pg�4�

P�n� � �1 � Pg�1�� �1 � Pg�2�� �1 � Pg�3����1 � Pg�i���

�1 � Pg�n � 1��Pg�n�or

P�n� � Pg�n��i�1n�1�1 � Pg�i�� (12)

When Pg(i), or Pg(t), is a known algebraic function, equation 12 can be

FIG 5 Comparison of generated dynamic germination curves for a quasi-isothermal temperature profile and corresponding truly isothermal curve, which hasbeen 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 forthe 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 generatedwith the two models are indistinguishable.

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used to estimate the probability of a spore’s being found germinated afterany time t, using an interpolation function for a noninteger time (18). Onecan also write an iterative algorithm that complements the Markov chainby casting the probabilities and checking whether the spore has germi-nated. Once it has, the iterations stop. The process can then be repeatedwith 10, 50 100, or more individual spores, or groups of spores, to generatethe germination curves of their respective populations and then comparethem with those described by the deterministic phenomenological models(in our case, equation 4 or 5 for the isothermal conditions). Or, in the caseof dynamic germination, at least in principle one can compare the resultingcurve produced by the stochastic model with the solution of equation 8 or 9.As in growth and inactivation (19–21), the stochastic model’s structure is thesame regardless of whether the temperature is constant or varies with time.The difference will be manifested in the probability rate function, Pg(t) orPg(i), which would be affected by the changing temperature, but not in themodel’s mathematical structure, which will remain the same.

RESULTS AND DISCUSSIONSimulating dynamic germination. Under dynamic conditions,the germination parameters Pasym(T), tc(T), and m(T), whosetemperature dependencies are probably of the kinds shown in Fig.4, become functions of time, i.e., Pasym(t) � Pasym(T(t)), tc(t) � tc

(T(t)), and m(t) � m(T(t)). These terms are incorporated into therate equation’s coefficients (those of equation 8 or 9), and theresulting differential equation is solved numerically. The solutionis the dynamic germination curve for the particular temperatureprofile T(t). Examples of nonisothermal curves generated as thesolutions of such rate equations are shown in Fig. 7 and 8. Thesesimulated germination curves are for hypothetical oscillating tem-perature profiles with a rising or falling trend. The temperatureprofiles, T(t)’s, are displayed at the top of each figure. Below eachtemperature profile is the corresponding germination curve, pre-sented as the percent germination as a function of time. Shown atthe bottom are plots of the germination curves’ derivatives, de-

picting the germination rate as percent per unit time as a functionof time. (In certain oscillating temperature profiles, when thegenerated germination curve approaches the asymptotic ger-mination level, the numerical solution of the rate equation canbecome a list of complex numbers from a certain point onward.When this happens, one has to truncate the curve and replaceits continuation by the value of the function just before a sig-nificant imaginary component appears. This value representsthe practically constant asymptotic germination level for theparticular temperature profile.) Again, the examples given areof scenarios where germination approaches 100%. The calcu-lation method, however, is the same for lower germinationlevels. A procedure for choosing the germination model type (A orB), three temperature profile kinds, monotonically rising or falling oroscillating temperature, and other parameters values has been auto-mated and posted on the Internet in the form of a freely download-able Wolfram demonstration entitled “Dynamic Germination ofSeeds and Microbial Spores” (http://demonstrations.wolfram.com/DynamicGerminationOfSeedsAndMicrobialSpores/). The screendisplay of this demonstration is shown in Fig. 9. Apart from en-abling the user to simulate germination patterns with parametersvarying within their permitted ranges, the demonstration allowsfree access to the program’s code. Thus, users who have Mathematicacan modify or replace the germination models, create alternativetemperature profiles, and/or change the ranges of Pasym(T), tc(T), andm(T) or their equivalents in alternative models.

To confirm the described germination model and calcula-tion procedure, one should generate nonisothermal germina-tion curves with parameters obtained from isothermal experi-mental data and test the predictions against actual curvesobtained under the corresponding temperature profiles. Pass-ing this test will validate the model and the assumptions on

FIG 6 Probabilities tree (Markov chain) of a single germinating dormant spore.

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which it is based. Once validated, the method could be used topredict germination patterns under a variety of real and hypo-thetical dynamic conditions and examine their potential con-sequences or implications.

Simulating germination patterns with the stochastic model.Examples of raw germination curves generated with the fully sto-chastic model (equation 12) are given in Fig. 10. (We use the term“raw” here, since the plots depict the total number of germinated spores

versus timeandnot theirpercentage.)Thecurveswereallproducedwiththe same underlying sigmoid probability rate function:

Pg�t� �Pg0

1 � exp�kg�tg � t�� (13)

where Pg0 is the asymptotic level, kg is a constant, and tg is a char-acteristic time.

Figure 10 shows the germination pattern of initially dormant

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.

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spores in assemblies of 10, 30, 100, and 500. As expected (20, 21),as the group size rises, the assembly’s germination curve becomessmoother and more deterministic. The role of the germinationprobability rate function is revealed in Fig. 11. It demonstratesthat when Pg(t) is itself sigmoid, following equation 13, for exam-

ple, so is the germination curve, i.e., as produced by equation 4, 5,7, or 8 with m � 1. When Pg(t) is a constant, Pg(t) � constant �100%, or a monotonically declining function, the germinationcurve would be nonsigmoid, having a shape of the kind producedby the deterministic phenomenological model A or B 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.

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As also shown in Fig. 11, peaked Pg(t), here produced with theequation

Pg�t� �Pg0

1 � exp�kg�tg1 � t��exp��� t

tg2�n (14)

where tg1, tg2, and n are constants, can produce germination curveswhose asymptotic level depends on the peak’s height and location.In other words, the asymptotic germination level, represented byPasym in equations 4 to 11, would not exceed Pg0 in equation 14,i.e., Pasym � Pg0.

The stochastic model presented in Fig. 6 and generated withequations 12 to 14 is in fact a special case of a more general model,which would need to account for simultaneous spore inactivationand cell division and mortality, as shown schematically in Fig. 12.In the simplest model that can describe such a situation, the ger-mination probability rate function, Pg(t), would have to be ac-companied by the spores’ inactivation probability rate functionand the probability rate functions of the vegetative cells’ mortalityand of their division, all determined by the organism, its temper-ature history, and the medium in which these processes occur. Wesay the “simplest” because to accurately account for spore inacti-vation and cell mortality and division, one might need to considerdifferent probability rate functions for the different branches of

the probabilities tree (21). Discussion and mathematical descrip-tion of such general scenarios is outside the scope of the presentwork. However, one can argue that even in its simplified version,the presented model (equation 12) can still provide a qualitativeexplanation of the emergence of certain germination patterns,where inactivation and cell mortality and division do not play amajor role on the pertinent time scale.

Concluding remarks. At least in principle, it is possible to con-vert models used to describe microbial germination under iso-thermal conditions into differential rate equations applicable todynamic conditions. The resulting models, although cumbersomelooking, are ordinary differential equations that can be easily andrapidly solved with commercial software. This is demonstrated ina freely downloadable interactive mathematical program whichhas been posted on the Internet. The described rate models havepassed the test of consistency. However, only future research willshow whether these models can be used not only to simulate dy-namic germination patterns but also to predict them from exper-imental isothermal or other dynamic data, as has been done withmodels of microbial inactivation and growth, which had beenderived from similar assumptions. When unaccompanied byspore inactivation, cell division, and mortality, the germination

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 themodel parameters can be entered and varied with sliders.

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process can be described by a simple Markov chain (i.e., a proba-bilities tree). The resulting purely stochastic model can be used togenerate the germination curves of individual and small groups ofspores of the kind that might be encountered in certain situations

of food safety, public health, and medical interest. As expected,when the number of germinating spores is increased, the germi-nation curve becomes smoother and the pattern more determin-istic.

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 thenumber 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 thesuperimposed solid gray curve is the fit of model A (equation 4). Notice that a constant (or falling) probability rate produces a nonsigmoid germination curve.

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The stochastic model is based on probability rates that varywith time. The model shows that if the probability rate functionrises monotonically, the germination curve assumes the com-monly observed sigmoid shape with an asymptote determined bythe spore type, temperature, and medium. When the probabilityrate is constant, the germination curve is not sigmoid but climbs ata decreasing rate, as has been observed in certain bacillus sporesunder high hydrostatic pressure. If the probability rate rises to apeak value and then falls, the germination curve can still be sig-moid. The asymptotic germination level in this case would dependon the peak’s location and height. Although not attempted in thiswork, it is possible to determine the probability rate function fromgermination data obtained with large spore populations (20, 21).Once determined, this probability function could be used to sim-ulate and examine the germination patterns of small groups ofspores, and even individual ones, under circumstances relevant tofood safety or infection.

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FIG 12 Germination probabilities tree when cell division and mortality occursimultaneously.

Static and Dynamic Spore Germination

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