Fit a four-parameter hill function (El-Kassaby et al. 2008) to cumulative germination count data and compute the associated parameters.
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
)Germination counts at each time interval. Can be partial
or cumulative as specified in the argument partial.
The time intervals.
Total number of seeds.
logical. If TRUE, germ.counts is considered as
partial and if FALSE, it is considered as cumulative. Default is
TRUE.
Force the intercept of the y axis through 0.
Fix a as the actual maximum germination percentage at the end of the experiment.
The time up to which AUC is to be computed.
Germination percentage value(s) for which the corresponding time is
to be computed as a numeric vector. Default is c(10, 60).
The minimum germination percentage value for computing
uniformity. Default is 10. Seed Details.
The maximum germination percentage value for computing
uniformity. Default is 90. Seed Details.
The number of tries to be attempted to fit the curve. Default is 3.
A list with the following components:
A data frame with the data used for computing the model.
A data frame of parameter estimates, standard errors and p value.
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.
The asymptote or the maximum cumulative germination percentage.
The mathematical parameter controlling the shape and steepness of the germination curve.
The half-maximal activation level.
The intercept on the y axis.
Time at germination onset.
duration between the time at germination onset (lag) and that at 50% germination.
Time required for 50% of total seeds to germinate. Will
be NaN if more than 50% of total seeds do not germinate.
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.
Time required for 50% of viable/germinated seeds to germinate.
Time
required for x% (as specified in argument xp) of viable/germinated
seeds to germinate.
Time interval between umin%
and umax% of viable seeds to germinate.
Time at maximum germination rate.
The estimate of area under the curve.
Mean germination time.
Skewness of mean germination time.
The message from
gsl_nls.
Logical value indicating whether convergence was achieved.
The raw fitted model output as a list of class
gsl_nls.
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.
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.
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)
#> $data
#> gp csgp intervals
#> 1 0 0 1
#> 2 0 0 2
#> 3 0 0 3
#> 4 0 0 4
#> 5 8 8 5
#> 6 34 42 6
#> 7 20 62 7
#> 8 14 76 8
#> 9 2 78 9
#> 10 0 78 10
#> 11 2 80 11
#> 12 0 80 12
#> 13 0 80 13
#> 14 0 80 14
#>
#> $Parameters
#> term estimate std.error statistic p.value
#> 1 bta 2.290709 0.05602634 40.88628 2.965932e-14
#> 2 c 6.034954 0.03872162 155.85488 3.270089e-21
#>
#> $Fit
#> sigma isConv finTol logLik AIC BIC deviance df.residual
#> 1 1.61522 TRUE 2.852829e-12 -25.49868 56.99736 58.91453 31.30723 12
#> nobs
#> 1 14
#>
#> $a
#> [1] 80
#>
#> $b
#> [1] 9.881937
#>
#> $c
#> [1] 6.034954
#>
#> $y0
#> [1] 0
#>
#> $lag
#> [1] 0
#>
#> $Dlag50
#> [1] 6.034954
#>
#> $t50.total
#> [1] 6.355121
#>
#> $txp.total
#> 10 60
#> 4.956264 6.744598
#>
#> $t50.Germinated
#> [1] 6.034954
#>
#> $txp.Germinated
#> 10 60
#> 4.831807 6.287724
#>
#> $Uniformity
#> 90 10 uniformity
#> 7.537690 4.831807 2.705882
#>
#> $TMGR
#> [1] 5.912194
#>
#> $AUC
#> [1] 1108.976
#>
#> $MGT
#> [1] 6.632252
#>
#> $Skewness
#> [1] 1.098973
#>
#> $msg
#> [1] "#1. success "
#>
#> $isConv
#> [1] TRUE
#>
#> $model
#> Nonlinear regression model
#> model: csgp ~ FourPHF_fixa_fixy0(x = intervals, a = max(csgp), bta, c)
#> data: data
#> bta c
#> 2.291 6.035
#> residual sum-of-squares: 31.31
#>
#> Algorithm: multifit/levenberg-marquardt, (scaling: levenberg, solver: qr)
#>
#> Number of iterations to convergence: 8
#> Achieved convergence tolerance: 2.853e-12
#>
#> attr(,"class")
#> [1] "FourPHFfit" "list"
# From cumulative germination counts
#----------------------------------------------------------------------------
FourPHFfit(germ.counts = y, intervals = int, total.seeds = 50, tmax = 20,
partial = FALSE)
#> $data
#> gp csgp intervals
#> 1 0 0 1
#> 2 0 0 2
#> 3 0 0 3
#> 4 0 0 4
#> 5 8 8 5
#> 6 34 42 6
#> 7 20 62 7
#> 8 14 76 8
#> 9 2 78 9
#> 10 0 78 10
#> 11 2 80 11
#> 12 0 80 12
#> 13 0 80 13
#> 14 0 80 14
#>
#> $Parameters
#> term estimate std.error statistic p.value
#> 1 bta 2.290709 0.05602634 40.88628 2.965932e-14
#> 2 c 6.034954 0.03872162 155.85488 3.270089e-21
#>
#> $Fit
#> sigma isConv finTol logLik AIC BIC deviance df.residual
#> 1 1.61522 TRUE 2.852829e-12 -25.49868 56.99736 58.91453 31.30723 12
#> nobs
#> 1 14
#>
#> $a
#> [1] 80
#>
#> $b
#> [1] 9.881937
#>
#> $c
#> [1] 6.034954
#>
#> $y0
#> [1] 0
#>
#> $lag
#> [1] 0
#>
#> $Dlag50
#> [1] 6.034954
#>
#> $t50.total
#> [1] 6.355121
#>
#> $txp.total
#> 10 60
#> 4.956264 6.744598
#>
#> $t50.Germinated
#> [1] 6.034954
#>
#> $txp.Germinated
#> 10 60
#> 4.831807 6.287724
#>
#> $Uniformity
#> 90 10 uniformity
#> 7.537690 4.831807 2.705882
#>
#> $TMGR
#> [1] 5.912194
#>
#> $AUC
#> [1] 1108.976
#>
#> $MGT
#> [1] 6.632252
#>
#> $Skewness
#> [1] 1.098973
#>
#> $msg
#> [1] "#1. success "
#>
#> $isConv
#> [1] TRUE
#>
#> $model
#> Nonlinear regression model
#> model: csgp ~ FourPHF_fixa_fixy0(x = intervals, a = max(csgp), bta, c)
#> data: data
#> bta c
#> 2.291 6.035
#> residual sum-of-squares: 31.31
#>
#> Algorithm: multifit/levenberg-marquardt, (scaling: levenberg, solver: qr)
#>
#> Number of iterations to convergence: 8
#> Achieved convergence tolerance: 2.853e-12
#>
#> attr(,"class")
#> [1] "FourPHFfit" "list"