The biogrowth package implements two modeling
approaches. The first one is based on the use of primary growth models
to describe the relationship between the population size and the elapsed
time. The second one, expands the primary model considering the effect
of changes in the experimental conditions on the specific growth rate
(parameter \(\mu\)). This is reflected
in the argument environment of
predict_growth() (and also fit_growth()),
which can take two values: “constant” or “dynamic”. In the first case,
the function only takes a primary_model (indeed, passing a
secondary_model will return a warning). The logic for this
is that the secondary model describes how the environmental conditions
affect the kinetic parameters of the primary model (the growth rate).
Therefore, the minimum model to describe growth under static conditions
is the one that only includes a primary model. Nonetheless, the function
can still predict microbial growth under isothermal conditions using
environment="dynamic". This can be useful in situations
were the environmental conditions are constant, but the response of the
population is described using secondary models.
To show this, let’s define an environmental profile where we only consider a constant temperature of 35ºC.
Next, we define primary and secondary models as usual.
q0 <- 1e-4
mu_opt <- .5
my_primary <- list(mu_opt = mu_opt,
Nmax = 1e8,N0 = 1e2,
Q0 = q0)
sec_temperature <- list(model = "CPM",
xmin = 5, xopt = 35, xmax = 40, n = 2)
my_secondary <- list(temperature = sec_temperature)Finally, we call predict_growth after defining the time
points of the simulation.
my_times <- seq(0, 50, length = 1000)
## Do the simulation
dynamic_prediction <- predict_growth(environment = "dynamic",
my_times,
my_primary,
my_secondary,
my_conditions)Because the temperature during the simulation equals the cardinal
parameter \(X_{opt}\), the predicted
population size is identical to the one calculated using
predict_growth with environment="constant" for
the Baranyi model (calculations for environment="dynamic"
are always based on the Baranyi model) when \(\mu = \mu_{opt}\) and \(\lambda = \frac{ \ln \left(1 +1/Q_0 \right)
}{\mu_{opt}}\).
lambda <- Q0_to_lambda(q0, mu_opt)
primary_model <- list(model = "Baranyi",
logN0 = 2, logNmax = 8, mu = mu_opt, lambda = lambda)
static_prediction <- predict_growth(my_times, primary_model)
plot(static_prediction) +
geom_line(aes(x = time, y = logN), linetype = 2,
data = dynamic_prediction$simulation,
colour = "green")
The advantages of using a model including a secondary model for
modeling growth under constant environmental conditions are evident when
simulations are made for several temperatures. Using
environment="constant" would require a calculation of the
value of \(\mu\) for each temperature
separately. Because the relationship between \(\mu\) and temperature is included in the
secondary model, a separate calculation is not required when using
environment="dynamic".
max_time <- 100
c(15, 20, 25, 30, 35) %>% # Temperatures for the calculation
set_names(., .) %>%
map(., # Definition of constant temperature profile
~ data.frame(time = c(0, max_time),
temperature = c(., .))
) %>%
map(., # Growth simulation for each temperature
~ predict_growth(environment = "dynamic",
my_times,
my_primary,
my_secondary,
env_conditions = .,
logbase_mu = 10)
) %>%
imap_dfr(., # Extract the simulation
~ mutate(.x$simulation, temperature = .y)
) %>%
ggplot() +
geom_line(aes(x = time, y = logN, colour = temperature)) +
theme_cowplot()
Note, however, that predict_growth() does not include
any secondary model for the lag phase. The reason for this is that there
are no broadly accepted secondary models for the lag phase in predictive
microbiology. Therefore, the value of \(\lambda\) varies among the simulations
according to \(\lambda(T) = \frac{ \ln \left(1
+1/Q_0 \right) }{\mu(T)}\).
Another application of predict_growth() with
environment="dynamic" is including the impact of another
environmental factor when temperature is kept constant. This can be done
by defining a second secondary model.
my_primary <- list(mu_opt = mu_opt,
Nmax = 1e8,N0 = 1e2,
Q0 = q0)
sec_temperature <- list(model = "CPM",
xmin = 5, xopt = 35, xmax = 40, n = 2)
sec_pH <- list(model = "CPM",
xmin = 4, xopt = 7, xmax = 8, n = 2)
my_secondary_2 <- list(temperature = sec_temperature,
pH = sec_pH)Then, we can call predict_growth().
max_time <- 100
c(5, 5.5, 6, 6.5, 7, 7.5) %>% # pH values for the calculation
set_names(., .) %>%
map(., # Definition of constant temperature profile
~ tibble(time = c(0, max_time),
temperature = c(35, 35),
pH = c(., .))
) %>%
map(., # Growth simulation for each temperature
~ predict_growth(environment = "dynamic",
my_times,
my_primary,
my_secondary_2,
env_conditions = .,
logbase_mu = 10)
) %>%
imap_dfr(., # Extract the simulation
~ mutate(.x$simulation, pH = .y)
) %>%
ggplot() +
geom_line(aes(x = time, y = logN, colour = pH)) +
theme_cowplot()
As above, note that the lag phase varies between the simulations according to \(\lambda(T, pH) = \frac{ \ln \left(1 +1/Q_0 \right) }{\mu(T, pH)}\).