Description Usage Details Value Author(s) References Examples

Formulas of secondary growth models commonly used in predictive microbiology

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All the models describe the evolution of the square root of the maximum specific growth rate (sqrtmumax) as a function of one or more environmental factors among temperature (T), pH (pH) and water activity (aw). Each model must be fitted to a data frame including at least two columns, the last one named "sqrtmumax" and the first ones named "T", "pH" or "aw" according to the model.

`cpm_T`

is the cardinal temperature model with inflection (Rosso et al., 1993) with four parameters (Tmin, Topt, Tmax,muopt)

`cpm_pH_4p`

is the cardinal pH model (Rosso et al., 1995) with four parameters (pHmin, pHopt, pHmax, muopt)

`cpm_pH_3p`

is a symetric cardinal pH model with three parameters (pHmin, pHopt, muopt), obtained by fixing pHmax to 2pHopt-pHmin in the `cpm_pH_4p`

model

`cpm_aw_3p`

is the cardinal aw model (Rosso and Robinson, 2001) with three parameters (awmin, awopt, muopt)

`cpm_aw_2p`

is a simplified cardinal aw model (Rosso and Robinson, 2001) with two parameters (awmin, muopt) obtained by fixing awopt to 1 in the `cpm_aw_3p`

model

`cpm_T_pH_aw`

is the cardinal model based on the gamma concept (Pinon et al., 2004) with 9 parameters (Tmin, Topt, Tmax, pHmin, pHopt, pHmax, awmin, awopt, muopt)

A formula

Florent Baty [email protected]

Marie-Laure Delignette-Muller [email protected]

Pinon A, Zwietering M, Perrier L, Membr\'e J, Leporq B, Mettler E, Thuault D, Coroller L, Stahl V, Vialette M (2004) Development and validation of experimental protocols for use of cardinal models for prediction of microorganism growth in food products. *Applied Environmental Microbiology*, **70**, 1081-1087.

Rosso L, Robinson TP (2001) A cardinal model to describe the effect of water activity on the growth of moulds. *International Journal of Food Microbiology*, **63**, 265-273.

Rosso L, Lobry JR, Bajard S, Flandrois JP (1995) Convenient model to describe the combined effects of temperature and pH on microbial growth. *Applied Environmental Microbiology*, **61**, 610-616.

Rosso L, Lobry JR, Flandrois JP (1993) An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model. *Journal of Theoretical Biology* **162**, 447-463.

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data(ross)
# Example for the cpm_T model
d1 <- subset(ross, author == "salter" & aw == 0.997,
select = c(T, sqrtmumax))
nls1 <- nls(cpm_T, d1, list(muopt = 1.7, Tmin = 4, Topt = 40, Tmax = 47))
plotfit(nls1, smooth = TRUE)
overview(nls1)
# Example for the cpm_pH_4p model
d2 <- subset(ross, author == "presser" & aw > 0.99,
select = c(pH, sqrtmumax))
nls2 <- nls(cpm_pH_4p, d2, list(muopt = 0.5, pHmin = 4,
pHopt = 6.5, pHmax = 9))
plotfit(nls2, smooth = TRUE)
overview(nls2)
# Example for the cpm_pH_3p model
d3 <- subset(ross, author == "presser" & aw == 0.997,
select = c(pH, sqrtmumax))
nls3 <- nls(cpm_pH_3p, d3, list(muopt = 0.5, pHmin = 4, pHopt = 6.5))
plotfit(nls3, smooth = TRUE)
overview(nls3)
# Example for the cpm_aw_3p model
d4<-subset(ross, author == "mellefont", select = c(aw, sqrtmumax))
nls4 <- nls(cpm_aw_3p, d4, list(muopt = 0.6, awmin = 0.95, awopt = 0.995))
plotfit(nls4, smooth = TRUE)
overview(nls4)
# Example for the cpm_aw_2p model
d5 <- subset(ross, author == "mellefont" & aw < 0.99,
select = c(aw, sqrtmumax))
nls5 <- nls(cpm_aw_2p, d5, list(muopt = 0.6, awmin = 0.95))
plotfit(nls5, smooth = TRUE)
overview(nls5)
# Examples for the cpm_T_pH_aw model
d6 <- subset(ross, select = c(T, pH, aw, sqrtmumax))
nls6 <- nls(cpm_T_pH_aw, d6, list(muopt = 2, Tmin = 4, Topt = 40, Tmax = 49,
pHmin = 4, pHopt = 6.5, pHmax = 9, awmin = 0.95, awopt = 0.995))
def.par <- par(no.readonly = TRUE)
par(mfrow = c(2, 2))
plotfit(nls6, variable = 1)
plotfit(nls6, variable = 2)
plotfit(nls6, variable = 3)
overview(nls6)
par(def.par)
``` |

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