FourPHFfit {germinationmetrics}

Fit four-parameter hill function

Description

Fit a four-parameter hill function (El-Kassaby et al. 2008) to cumulative germination count data and compute the associated parameters.

Usage

FourPHFfit(
  germ.counts,
  intervals,
  total.seeds,
  partial = TRUE,
  fix.y0 = TRUE,
  fix.a = TRUE,
  tmax,
  xp = c(10, 60),
  umin = 10,
  umax = 90,
  tries = 3
)

Arguments

germ.counts

Germination counts at each time interval. Can be partial or cumulative as specified in the argument partial.

intervals

The time intervals.

total.seeds

Total number of seeds.

partial

logical. If TRUE, germ.counts is considered as partial and if FALSE, it is considered as cumulative. Default is TRUE.

fix.y0

Force the intercept of the y axis through 0.

fix.a

Fix a as the actual maximum germination percentage at the end of the experiment.

tmax

The time up to which AUC is to be computed.

xp

Germination percentage value(s) for which the corresponding time is to be computed as a numeric vector. Default is c(10, 60).

umin

The minimum germination percentage value for computing uniformity. Default is 10. Seed Details.

umax

The maximum germination percentage value for computing uniformity. Default is 90. Seed Details.

tries

The number of tries to be attempted to fit the curve. Default is 3.

Details

The cumulative germination count data of a seed lot can be modelled to fit a four-parameter hill function defined as follows (El-Kassaby et al. 2008).

\[y = y_{0}+\frac{ax^{b}}{c^{b}+x^{b}}\]

Where, \(y\) is the cumulative germination percentage at time \(x\), \(y_{0}\) is the intercept on the y axis, \(a\) is the asymptote, or maximum cumulative germination percentage, which is equivalent to germination capacity, \(b\) is a mathematical parameter controlling the shape and steepness of the germination curve (the larger the \(b\) parameter, the steeper the rise toward the asymptote \(a\), and the shorter the time between germination onset and maximum germination), and \(c\) is the "half-maximal activation level" which represents the time required for 50% of viable seeds to germinate (\(c\) is equivalent to the germination speed).

In FourPHFfit, this model has be reparameterized by substituting \(b\) with \(e^{\beta}\) to constraint \(b\) to positive values only.

\[y = y_{0}+\frac{ax^{e^{\beta}}}{c^{e^{\beta}}+x^{e^{\beta}}}\]

Where, \(b = e^{\beta}\) and \(\beta = \log_{e}(b)\).

The curve fitting is performed with nonlinear gslnls package, a R interface to the least-squares optimization with the GNU Scientific Library (GSL) with the Levenberg-Marquardt algorithm (Chau 2023).

Once this function is fitted to the curve, FourPHFfit computes the time to 50% germination of total seeds (t50.total) or viable seeds (t50.Germinated). Similarly the time at any percentage of germination (in terms of both total and viable seeds) as specified in argument xp can be computed.

The time at germination onset (\(lag\)) can be computed as follows.

\[lag = b\sqrt{\frac{-y_{0}c^{b}}{a + y_{0}}}\]

The value \(D_{lag-50}\) is defined as the duration between the time at germination onset (lag) and that at 50% germination (\(c\)).

The time interval between the percentages of viable seeds specified in the arguments umin and umin to germinate is computed as uniformity (\(U_{t_{max}-t_{min}}\)).

\[U_{t_{max}-t_{min}} = t_{max} - t_{min}\]

The partial derivative of the four-parameter hill function gives the instantaneous rate of germination (\(s\)) as follows.

\[s = \frac{\partial y}{\partial x} = \frac{abc^{b}x^{b-1}}{(c^{b}+x^{b})^{2}}\]

From this function for instantaneous rate of germination, the time at maximum germination rate (\(TMGR\)) can be estimated as follows.

\[TMGR = b \sqrt{\frac{c^{b}(b-1)}{b+1}}\]

TMGR represents the point in time when the instantaneous rate of germination starts to decline.

The area under the curve (\(AUC\)) is obined by integration of the fitted curve between time 0 and time specified in the argument 'tmax'.

Integration of the fitted curve gives the value of mean germination time (\(MGT\)) and the skewness of the germination curve is computed as the ratio of \(MGT\) and the time for 50% of viable seeds to germinate (\(t_{50}\)).

\[Skewness = \frac{MGT}{t_{50}}\]

If final germination percentage is less than 10%, a warning is given, as the results may not be informative.

Value

A list with the following components:

data

A data frame with the data used for computing the model.

Parameters

A data frame of parameter estimates, standard errors and p value.

Fit

A one-row data frame with estimates of model fitness such as log likelyhoods, Akaike Information Criterion, Bayesian Information Criterion, deviance and residual degrees of freedom.

a

The asymptote or the maximum cumulative germination percentage.

b

The mathematical parameter controlling the shape and steepness of the germination curve.

c

The half-maximal activation level.

y0

The intercept on the y axis.

lag

Time at germination onset.

Dlag50

duration between the time at germination onset (lag) and that at 50% germination.

t50.total

Time required for 50% of total seeds to germinate. Will be NaN if more than 50% of total seeds do not germinate.

txp.total

Time required for x% (as specified in argument xp) of total seeds to germinate. Will be NaN if more than x% of total seeds do not germinate.

t50.Germinated

Time required for 50% of viable/germinated seeds to germinate.

txp.Germinated

Time required for x% (as specified in argument xp) of viable/germinated seeds to germinate.

Uniformity

Time interval between umin% and umax% of viable seeds to germinate.

TMGR

Time at maximum germination rate.

AUC

The estimate of area under the curve.

MGT

Mean germination time.

Skewness

Skewness of mean germination time.

msg

The message from gsl_nls.

isConv

Logical value indicating whether convergence was achieved.

model

The raw fitted model output as a list of class gsl_nls.

References

Chau J (2023). “gslnls: GSL Nonlinear Least-Squares Fitting.”

El-Kassaby YA, Moss I, Kolotelo D, Stoehr M (2008). “Seed germination: Mathematical representation and parameters extraction.” Forest Science, 54(2), 220–227.

Galassi M (ed.) (2009). GNU Scientific Library Reference Manual: For GSL Version 1.12, 3. ed edition. Network Theory, Bristol. ISBN 978-0-9546120-7-8.

Examples

x <- c(0, 0, 0, 0, 4, 17, 10, 7, 1, 0, 1, 0, 0, 0)
y <- c(0, 0, 0, 0, 4, 21, 31, 38, 39, 39, 40, 40, 40, 40)
int <- 1:length(x)
total.seeds = 50

# From partial germination counts
#----------------------------------------------------------------------------
FourPHFfit(germ.counts = x, intervals = int, total.seeds = 50, tmax = 20)

# From cumulative germination counts
#----------------------------------------------------------------------------
FourPHFfit(germ.counts = y, intervals = int, total.seeds = 50, tmax = 20,
           partial = FALSE)