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Seed Germination: Mathematical Representation and Parameters
Extraction
Yousry A. El-Kassaby, Ian Moss, Dave Kolotelo, and Michael Stoehr
Abstract: A method for mathematically describing cumulative seed germination using the four-parameter Hill
function is described. The function’s four parameters allowed both direct or indirect biological interpretation of
germination behavior and the impact of seed pretreatments on germination improvement. Three parameters, a,
b, and c, allowed direct assessment of germination capacity (%), the shape and steepness of the germination
course, and germination speed (time to reach 50% germination). The fourth parameter y
0
permitted estimation
of the lag time (germination onset). The mathematical expression of cumulative seed germination of unstratified
and stratified seed made it possible to quantitatively estimate seedlot dormancy and the amount of germination
improvement caused by a specific seed pretreatment. This was accomplished by estimating the area between the
two cumulative germination curves through integration. The utility of the proposed approach was demonstrated
using germination data from wind-pollinated, individual genotypes (half-sib families) of lodgepole pine (Pinus
contorta var. latifolia) and bulk seedlots of lodgepole pine and white spruce (Picea glauca). FOR.SCI. 54(2):
220 –227.
Keywords: seed germination, germination parameters, dormancy, four-parameter Hill function, curve-fitting
THE GERMINATION PERFORMANCE of a seedlot can be
characterized by three parameters; time of germina-
tion onset (lag), germination speed (rate), and extent
or capacity (cumulative germination percentage at the end
of the testing period). The genetic makeup of a seedlot
varies from a bulk collection (e.g., originating from an
unknown number of parents) to a full-sib family (derived
from controlled pollination) and strongly affects the vari-
ability of its germination parameters (El-Kassaby et al.
2002). Germination parameters are useful for estimating the
conversion of seeds to seedlings and, thus, the suitability of
a seedlot for commercial seedling production. Germination
parameters are also useful in determining the type of seed
pretreatment as well as nursery management practices
needed to attain a high level of germination (Kolotelo et al.
2001).
In most cases, germination capacity (% germination) is
the most important parameter in determining the suitability
of a seedlot for commercial use, but germination rate influ-
ences the uniformity of emergence in nurseries (Ching
1959; Thomson and El-Kassaby 1993; El-Kassaby 2000).
Several attempts have been made to simplify the character-
ization of seed germination performance by distilling the
various germination parameters into a single index or value
(e.g., Czabator 1962). However, reducing multiple germi-
nation parameters into one index provides an incomplete
picture of germination behavior.
When germination is properly fitted to a mathematical
function, the parameters of that function can be used to
better understand germination behavior and the effect of
seed pretreatments on germination enhancement. Many
curve-fitting routines have been used and their suitabilities
for characterizing and managing seedlots have been criti-
cally assessed (Tipton 1984; Brown and Mayer 1988a,
1988b).
Curve-fitting methods using the Weibull, Gompertz, and
probit functions have been used to characterize the germi-
nation of conifer seeds, compare seed pretreatments, and
measure seedlot differences (Bonner and Dell 1976; Rink et
al. 1979; Campbell and Sorensen 1979; Bramlett et al. 1983;
Dell et al. 1983; Leadem 1986, Stoehr et al. 1998). How-
ever, the parameters in these functions do not lend them-
selves to simple biological interpretation. The use of curve-
fitting to interpret and understand germination patterns has
been reviewed and endorsed by Scott et al. (1984).
In this article, we introduce the four-parameter Hill func-
tion (4-PHF) as a curve-fitting method for describing conif-
erous seed germination. We also show that this function is
amenable to biological interpretation, thereby enhancing
our understanding of germination differences among geno-
types, seedlots, and seed pretreatments.
The 4-PHF and Germination Parameters
The cumulative germination count of a seedlot was mod-
eled to fit the 4-PHF (Equation 1) using the curve-fitting
routine in MATLAB (The MathWorks, Inc. 2005), with the
Yousry A. El-Kassaby, University of British Columbia, Forest Sciences, 2424 Main Mall, Vancouver, BC V6T 1Z4, Canada—Phone: (604) 822-1821; Fax:
(604) 822-9102; y.el-kassaby@ubc.ca. Ian Moss, ForesTree Dynamics Ltd.—forestree@shaw.ca. David Kolotelo, British Columbia Ministry of Forests and
Range, Tree Improvement Branch, Tree Seed Centre— dave.kolotelo@gov.bc.ca. Michael Stoehr, British Columbia Ministry of Forests and Range, Research
Branch—michael.stoehr@gov.bc.ca.
Acknowledgments: We extend our thanks to M. El-Sharkawi (University of Washington, Seattle, WA), M. Fayed and M. Yasein (University of Victoria,
Victoria, BC, Canada), and M. Senousy (University of British Columbia, Vancouver, BC, Canada) for assisting with the model development and introduction
to MATLAB. This work was funded, in part, by the Johnson’s Family Forest Biotechnology Fund, British Columbia Forest Genetics Council’s Applied Forest
Genetics and Biotechnology Grant, and The Natural Sciences and Engineering Research Council of Canada-Industry Research Chair Funds to Y.E.K.
Manuscript received March 14, 2007, accepted September 21, 2007 Copyright © 2008 by the Society of American Foresters
220 Forest Science 54(2) 2008
duration of the germination test and cumulative germination
percentage representing the x- and y-variables, respectively:
y⫽y0⫹axb
cb⫹xb, (1)
where yis the cumulative germination percentage at time x,
y
0
is the intercept on the yaxis (ⱕ0), ais the asymptote, or
maximum cumulative germination percentage, which is
equivalent to germination capacity, bis a mathematical
parameter controlling the shape and steepness of the germi-
nation curve (the larger the bparameter, the steeper the rise
toward the asymptote a, and the shorter the time between
germination onset and maximum germination), and cis the
“half-maximal activation level” measured in days (Keshet
2006) and represents the time required for 50% of viable
seeds to germinate (cis equivalent to the germination speed
[R
50
⬘] parameter of Thomson and El-Kassaby [1993]), and
xis time in days. The time at germination onset (lag) is
computed by solving Equation 1 after setting yto0as
follows:
lag ⫽b
冑
⫺y0cb
a⫹y0
, (2)
where y
0
is the intercept on the yaxis.
Another way to characterize the speed of germination is
to consider the duration (D
lag-50
) between the time at ger-
mination onset (lag) and that at 50% germination (c). The
shorter the D
lag-50
, the more prompt and uniform the ger-
mination and the steeper the germination curve (i.e., higher
b). Thus, D
lag-50
and bshould be negatively correlated, and
D
lag-50
is a good indictor of germination speed, uniformity,
and vigor. Collectively, the D
lag-50
,c, and lag parameters
provide a better indicator of germination speed than R
50
⬘
alone. This is because R
50
⬘is a function of time only and
could be the same for two seedlots with different germina-
tion behaviors (i.e., similar R
50
⬘but different germination
onset times and/or different germination capacities).
The instantaneous rate of germination was also estimated
from the partial derivative of Equation 1 and can be used to
measure the sensitivity of any specific variable as follows:
s⫽⭸y
⭸x⫽abcbxb⫺1
共cb⫹xb兲2, (3)
where sis the daily rate of germination. This function was
plotted against time, and the time required to reach maxi-
mum germination rate (time at maximum germination rate
[TMGR]) was determined as
TMGR ⫽b
冑
cb(b⫺1)
b⫹1, (4)
The location of TMGR on the germination curve is the point
at which the instantaneous slope is at maximum (Figure 1).
This point is different from Czabator’s “peak value (PV)”
which is “the maximum quotient derived from all of the
cumulative full-seed germination percent on any day di-
vided by the number of days to reach this percent” (Czaba-
tor 1962). The difference between PV and TMGR is similar
to the difference between mean germination increment and
periodic germination increment. The former represents the
maximum total amount of germination divided by the total
time elapsed, whereas the latter are maximum rates of
germination at a point or small interval in time that may be
alternatively referred to as instantaneous rates of germina-
tion. By definition the maximum instantaneous rate must
exceed the maximum average rate of germination and so
too, the maximum instantaneous rate must occur in advance
of the maximum average rate of germination (Table 1). Both
TMGR and PV are important to the production of seedlings.
However, TMGR defines the inflection point of the cumu-
lative germination curve or the point in time when the
instantaneous rate of germination starts to decline. It also
defines the scale of the cumulative germination curve as this
is proportional to the maximum instantaneous rate of ger-
mination. In contrast, PV is no more important than any
other point along the line in terms of uniquely defining the
shape and scale of a sigmoid curve. Thus, TMGR is bio-
logically more meaningful. The shorter the TMGR, the
more vigorous is the seedlot and the time to reach R
50
⬘is
shorter. Therefore, the 4-PHF is suitable for extracting
meaningful germination parameters and characterizing the
behavior of germination at different times during the ger-
mination test. Furthermore, if germination is expressed by a
mathematical function, then this function could be used to
assess the effectiveness of seed pretreatments using the
integration method of Richter and Switzer (1982). Using
this method, dormancy is defined as the amount of ger-
mination improvement caused by a specific dormancy-
breaking treatment. The improvement is quantitatively
determined by the difference between the areas under the
germination curves before and after seed pretreatment
(Figure 2) and is defined as the dormancy index (DI), which
is calculated as
DI ⫽兰t0
tn共y1⫺y2)dt, (5)
0 2 4 6 8 10 12 14 16 18 20
0
10
20
30
40
50
60
70
80
90
100
Days
Germination %
1st Derivative
0
5
10
15
20
25
30
35
40
45
50
Figure 1. The 4-PHF for stratified (solid line: upper curve) and un-
stratified (solid line: lower curve) seedlot of lodgepole pine (single
family) and the instantaneous rate of germination for the same strat-
ified (dashed line: upper curve) and unstratified (dashed solid line:
lower curve) seedlot. TMGR is the peak point for lower curves. The
dots represent the cumulative germination over time, whereas the solid
line represents the fitted curves.
Forest Science 54(2) 2008 221
where DI is the dormancy index (surface area), y
1
and y
2
are
the 4-PHF for the seedlot after and before the dormancy
breaking treatment, respectively, and t
0
and t
n
are the time
in days at the beginning and end of the germination test,
respectively. Thus, zero and negative values imply that the
treatment either has no effect on improving seedlot perfor-
mance, or is detrimental (see below).
Parameters Demonstration
Germination parameters for 18 wind-pollinated seedlots
(families) from individual lodgepole pine (Pinus contorta
var. latifolia) orchard parents were determined from stan-
dard germination protocols (International Seed Testing
Agency 2006). The seed pretreatment consisted of a simple
stratification treatment (soaked in water for 1 day) followed
by surface drying and storage at 4°C for 4 weeks. The seeds
were germinated at alternating temperatures between 30°C
for an 8-hour day and 20°C for a 16-hour night. Light, at
about 13.5
mol m
⫺2
s
⫺1
was provided during the day by
means of cool-white fluorescent tubes. Additionally, DI was
evaluated on bulk lodgepole pine and white spruce (Picea
glauca) seedlots under five stratification durations, ranging
from 1 to 5 weeks. Germination tests for individual family
and bulk seedlots were conducted on four replications of
100 seeds each. Germinant counts were conducted daily,
and the cumulative germination for each replication was
determined. A seed was considered as a germinant when the
emerging radicle was four times the length of the seed. The
germination test and seed count continued until the end of
the 21-day testing period.
Statistical Analyses
Germination parameters (a,b,c, TMGR, and lag) for
each replication within each family were generated by fit-
ting the 4-PHF curve and then subjecting the parameters to
analyses of variance using the additive linear model,
Yijk ⫽
⫹Ti⫹Fj⫹TFij ⫹
共ij兲k, (6)
where
is the overall mean, T
i
is the effect of the ith seed
pretreatment (stratification) (i⫽1 to 2, fixed effect), F
j
is
the effect of the jth family (i⫽1 to 18, random effect), TF
ij
is the effect of the interaction between ith family and seed
ith pretreatment; and
(ij)k
is the residual error (k⫽1to4).
With the exception of germination capacity (which was
Table 1. Comparison between Czabator’s peak values (PV) determined from cumulative germination over time and the Time at Maximum
Germination Rate (TMGR) determined from the instantaneous rate of germination values for the unstratified and stratified lodgepole pine seedlots
presented in Figure 1
Day
Unstratified Stratified
Cumulative
Germination PV TMGR
Cumulative
Germination PV TMGR
1 0 0.00 0.00 0 0.00 0.00
2 0 0.00 0.00 0 0.00 0.12
3 0 0.00 0.01 0 0.00 1.24
4 0 0.00 0.04 0 0.00 5.97
5 0 0.00 0.13 2 0.40 16.57
60 0.00 0.36 42 7.00 26.08
7 0 0.00 0.84 61 8.71 21.14
8 0 0.00 1.67 70 8.75 12.30
93 0.33 2.84 81 9.00 6.18
10 5 0.50 4.07 84 8.40 3.03
11 15 1.36 4.87 92 8.36 1.52
12 18 1.50 4.89 92 7.67 0.80
13 21 1.62 4.23 93 7.15 0.44
14 25 1.79 3.29 93 6.64 0.25
15 27 1.80 2.38 93 6.20 0.15
16 30 1.88 1.66 93 5.81 0.09
17 30 1.76 1.14 93 5.47 0.06
18 31 1.72 0.78 93 5.17 0.04
19 33 1.74 0.53 93 4.89 0.02
20 34 1.70 0.37 94 4.70 0.02
21 36 1.71 0.26 94 4.48 0.01
PV and TMGR values and their respective times are presented in bold.
Figure 2. DI is represented by the area between the germination
curves of stratified (upper curve) and unstratified (lower curve) for a
lodgepole pine seedlot (single family).
222 Forest Science 54(2) 2008
arcsine transformed), all parameters were analyzed using
their original values. Expected mean squares were calcu-
lated, and components of variance were estimated and their
percentage contributions to the total variation were
determined.
DI was analyzed after removing treatment effects from
the additive linear model given above (note that DI for each
replication was estimated using both stratified and unstrati-
fied germination tests, i.e., no treatment effect). Differences
among germination parameters for the stratification treat-
ments of the bulk lodgepole pine and white spruce seedlots
were presented graphically.
Results and Discussion
The reliance on a single germination index such as Cza-
bator’s germination value (Czabator 1962) or even any of its
individual components (i.e., mean daily germination or PV)
is inadequate. The reason for the inadequacy of these mea-
sures is their lack of association with time over the germi-
nation course, even though time is used to estimate each of
the index components. This caveat was recognized by seed
technologists, and PV values are commonly presented with
their respective time references (The British Columbia Min-
istry of Forests and Range’s Tree Seed Centre, pers,
comm.). Comparison of Czabator’s PV and TMGR for the
stratified and unstratified seedlots of lodgepole pine indi-
cated that germination behavior was not adequately de-
scribed by either parameter alone (Table 1 and Figure 1).
PV is almost meaningless unless it is reported with the
corresponding germination percentage. Similarly, TMGR
characterizes maximum daily germination but does not pro-
vide information on the germination value. Although both
values showed consistent trends for stratified and unstrati-
fied seed (i.e., higher PV value and lower TMGR, respec-
tively), their use alone does not allow one to draw any
informative conclusions about the germination behavior of a
seedlot. The peak values of 2 and 9 obtained for the un-
stratified and stratified germination courses; respectively,
are meaningless if they are not reported with the reference
times (i.e., 16 and 9 days) and the germination percentages
(i.e., 30 and 81%) (Table 1). Similarly, TMGR values of 12
and 6 days are meaningless if they are not reported with the
incremental increase in daily germination (Table 1). These
examples highlight the drawbacks of using a single “index”
or parameter.
Germination parameters derived by fitting the 4-PHF for
the 18 unstratified and stratified lodgepole pine families
produced several significant correlations (Table 2). For ex-
ample, there was a perfect correlation between TMGR and
R
50
⬘. Careful examination of TMGR and R
50
⬘values re-
vealed that they were almost equal, but R
50
⬘was consis-
tently slightly lower than TMGR. This indicates that TMGR
could be used as a reasonable proxy for R
50
⬘and vice versa.
This perfect correlation is not surprising. If cin Equation 1
is set to be equal to x, then
y⫽y0⫹acb
cb⫹cb⫽y0⫹a/2, (7)
Table 2. Person product-moment correlation coefficients among germination parameters for unstratified and stratified seedlots for 18 lodgepole
pine families (critical rⴝ0.232 and 0.302 for pat 5 and 1%, respectively; nⴝ18)
b*c⫽R
50
TMGR Lag D
lag-50
Unstratified
a0.41 ⫺0.25 ⫺0.22 ⫺0.02 ⫺0.24
b⫺0.75 ⫺0.70 0.19 ⫺0.82
c⫽R
50
1.00 0.18 0.91
TMGR 0.22 0.89
Lag ⫺0.24
Stratified
a0.18 ⫺0.21 ⫺0.21 ⫺0.13 ⫺0.18
b⫺0.43 ⫺0.37 0.27 ⫺0.71
c⫽R
50
1.00 0.63 0.89
TMGR 0.67 0.86
Lag 0.20
*See text for parameters description.
Table 3. ANOVA for the germination parameters of seedlots from 18 lodgepole pine families subjected to seed pretreatment (4 weeks stratification)
SOV df EMS abR
50
TMGR Lag D
Iag-50
DI
.............................%.............................
Treatment (T)* 1
2
e
⫹4
2
tf
⫹72
t
–** –** –** –** –** –** NA
†
Family (F)17
2
e
⫹8
2
f
36.4 37.6 58.9 57.0 22.9 43.9 93.4**
T⫻F17
2
e
⫹4
2
tf
52.0** 40.7** 32.3** 34.4** 18.8** 26.6** NA
Residual 108
2
e
11.6 21.7 8.8 8.6 58.3 30.5 6.6
SOV, source of variation; df, degrees of freedom; EMS, expected mean squares. See text for parameters description.
* Although no variance components or percent of total variation were estimated for the fixed effect (treatment), the treatment effect was highly significant
for all tested parameters.
†
DI was tested using a reduced additive linear model (i.e., no Tand T⫻Feffects) with residual’s df ⫽54.
** Significant at P⬍0.01.
Forest Science 54(2) 2008 223
which is precisely why the parameter cis referred to as
thehalf-maximum rate. Insofar as cdecreases, so too will
the time to the point of inflection because the inflection
point will generally be somewhere in the vicinity of the 50%
level of germination. Another correlation of interest was the
negative correlation between band D
lag-50
(⫺0.71 and
⫺0.82 for the stratified and unstratified seedlots, respec-
tively). This correlation confirms the role of germination
onset (promptness) and speed on compacting the germina-
tion duration, which in turn affects the curve shape and
steepness (Table 2). As expected, the faster the germination,
the earlier R
50
⬘is reached, resulting in a shorter D
lag-50
period. Thus, correlations between the time of germination
onset (lag) and both TMGR and R
50
⬘are expected to pro-
duce contrasting results for stratified (TMGR ⫽0.63, R
50
⬘
⫽0.67) and unstratified (TMGR ⫽0.18, R
50
⬘⫽0.22) seed.
These correlations indicate that the stratification treatment
was effective in speeding the onset of germination as well as
the time needed to reach 50% germination (Table 2). This
result is further confirmed by the high correlation between
TMGR and D
lag-50
for both stratified (0.86) and unstratified
(0.89) seed (Table 2). The parameter b(shape and steep-
ness) of the stratified seed curve did not correlate with its
germination capacity (a), indicating that the steepness of the
germination curve is not associated with higher germina-
tion. Finally as expected, bproduced significant negative
Dlag-50
1
2
3
4
5
Lag
3
4
5
6
Family
123456789101112131415161718
DI
10
20
30
40
50
60
Germination %
0
20
40
60
80
100
Family
123456789101112131415161718
R'50
4
6
8
10
12
b
5
10
15
20
Figure 3. Germination parameters for 18 stratified (E) and unstratified (F) lodgepole pine seedlots (families) (see text
for parameters explanation). Vertical lines represent 95% confidence intervals.
Stratified - unstratified germination %
0 20 40 60 80 100
DI
10
20
30
40
50
60
70
Figure 4. Correlation between the difference between stratified and
unstratified germination percentages versus DI as determined by the
Hill function for 18 lodgepole pine seedlots (families).
224 Forest Science 54(2) 2008
correlations with early germination parameters such as
TMGR and D
lag-50
(Table 2).
Variation in germination parameters was highly signifi-
cant for the treatment ⫻family interaction (Table 3). Thus,
graphical representations are best for interpreting treatment
and family effects (Figure 3). Seed pretreatment was highly
significant for all germination parameters; however, the use
of the mixed model did not allow us to estimate treatment
differences (i.e., quantifying fixed effects violates analysis
of variance assumptions). The family effect and the seed
pretreatment ⫻family interaction accounted for 23–59%
and 19 –52% of the variation, respectively (Table 3). These
results are in line with previously published estimates for
the same species (Krakowski and El-Kassaby 2005). How-
ever, the germination parameters tested in the present study
are the product of curve fitting using the 4-PHF. Graphical
0
20
40
60
80
100
US 1WK 2WK 3WK 4WK 5WK
Germ ination %
0
20
40
60
80
100
US 1WK 2WK 3WK 4WK 5WK
Germination %
0
1
2
3
4
5
6
US 1WK 2WK 3WK 4WK 5WK
Lag (Days )
0
1
2
3
4
5
US 1WK2WK3WK4WK5WK
Lag (Day s)
0
1
2
3
4
5
US 1WK 2WK 3WK 4WK 5WK
D
0-50
(Days)
0
1
2
3
4
5
US 1WK2WK3WK4WK5WK
D
0-50
(Days)
0
2
4
6
8
10
US 1WK2WK3WK4WK5WK
TMGR (Day )
0
2
4
6
8
10
US 1WK 2WK 3WK 4WK 5WK
TMGR (Day)
0
5
10
15
20
25
30
35
40
45
1WK 2WK 3WK 4WK 5WK
DI
0
5
10
15
20
25
30
35
40
45
1WK 2WK 3WK 4WK 5WK
DI
Figure 5. Interior spruce (left) and lodgepole pine (right) germination parameters as affected by stratification duration.
Black and white columns represent interior spruce seedlots 60751 and 60425 and lodgepole pine 2001 and 2002 seed
orchard lots, respectively. US, unstratified.
Forest Science 54(2) 2008 225
representation of each germination parameter allowed clear
assessment of either the seed pretreatment and family
effects on the observed variation (Figure 3). Seed pre-
treatment had a dominant effect on improving percent
germination (parameter a), lag, R
50
⬘, and D
lag-50
, result-
ing in smaller differences among families after stratifi-
cation (Figure 3). Parameter bconfirmed the observation
from the correlation analyses (above) and produced
mixed results for the stratified and unstratified seed. This
supports the hypothesis that the shape and steepness of
the germination curves are independent of germination
performance (Table 2 and Figure 3). Family differences
were large for all germination parameters, particularly for
unstratified seed; therefore, the stratification treatment
was successful in reducing these differences and harmo-
nizing germination among the 18 families (Figure 3).
DI varied significantly among families, with families
accounting for 93% of the total variation. This finding
indicates that dormancy level is a family-specific attribute
and is under strong genetic control (broad-sense
heritability/repeatability ⫽0.93) (Table 3 and Figure 3).
Although family dormancy differences were large, they
were substantially lessened by the stratification treatment,
producing uniform germination parameters (Figure 3). Ko-
lotelo (2006) advocated the development of a simple and
quick method for estimating dormancy so it could be used
operationally. Therefore, we determined the differences be-
tween the stratified and unstratified germination percent-
ages for the 18 lodgepole pine families and correlated these
differences with the actual DI for each family. Surprisingly,
a high and significant correlation (r⫽0.99) was obtained
(R
2
⫽0.978). This indicates that 98% of the variation in DI
could be explained by differences in percent germination
(Figure 4). Thus, we propose this approach for estimating
dormancy.
The lodgepole pine and white spruce bulk seedlots re-
sponded differently to the stratification treatments (Figure
5). Lodgepole pine bulk seedlots produced steady improve-
ment in percent germination with increasing stratification
time. This improvement was also associated with a steady
decline in the other parameters (lag, D
lag-50
, TMGR, and
R
50
⬘) (Figure 5). DI, estimated as the difference between the
areas under the various stratification treatments and the
control (unstratified), also showed a steady increase with
increasing stratification. An increase in DI indicates that the
difference between the stratified and unstratified germina-
tion curves is increasing and that stratification improves
germination. The trend observed for the white spruce bulk
seedlots mirrored that of the lodgepole pine seedlots (i.e.,
improved germination with stratification), but the two
spruce seedlots differed in their response (Figure 5). These
differences were most notable for D
lag-50
and DI, indicating
that extended stratification could have a detrimental effect
on white spruce. This finding indicates that generalizations
across the two species could be inappropriate. White spruce
seedlots responded similarly after 1 week of stratification,
and responses were inconsistent over the different stratifi-
cation times (Figure 5), suggesting that this species has
shallow dormancy. These results are consistent with earlier
work on a larger sample of 26 seedlots, including natural
stand and seed orchard collections (Kolotelo 1994).
When Hill (1910, 1913) first derived his sigmoidal func-
tion using three parameters, he based it on a mechanistic
model for the binding of oxygen to the enzyme hemoglobin
(Christopoulos and Lew 2000). The Hill equation was first
thought to be mechanistic because the resulting parameters
provided actual information about the underlying properties
of the interaction. Subsequent experiments revealed that it
was inadequate for this purpose, although it continues to be
used as a mechanistic model (i.e., when its validity is
confirmed) and as an empirical model when the shape of its
curve approximates that of experimental data.
The main reason for choosing the 4-PHF versus alterna-
tives (e.g., four-parameter Weibull, Gompertz, Chapman,
and logistic functions) was the ease with which the various
parameters could be interpreted. Equation 1 was also found
to fit the data with a reasonable level of precision, such that
reliable comparisons could be made between stratified and
unstratified seedlots. The program used to fit the equations
converged quickly onto each solution without the need to
arbitrarily constrain any of the parameters. For broad appli-
cation, however, it may be desirable to constrain the asymp-
tote to values ⱕ100% germination. Most of the fitted equa-
tions were statistically significant, and most of the R
2
values
were ⱖ0.9. A few seedlots with poor germination rates
(e.g., ⬍30% at the end of 21 days) produced lower Ftest
and R
2
values than those described above. In any event,
such seedlots would not be operationally deployed for seed-
ling production, particularly if they did not demonstrate
substantial improvement in germination after stratification
treatment.
No adjustments were made in the curve-fitting procedure
to deal with the effects of autocorrelation and heteroscedas-
ticity. Although TMGR is constrained to be less than c, this
is justified because TMGR tends to overestimate rather than
underestimate the inflection point. TMGR provides a better
estimate of the inflection point because it is based on all of
the data points. Finally, although it may have been possible
to obtain better fits using a different form of the sigmoidal
equation and better accounting of the error structure, this
would not have made any substantive difference to the
results or conclusions derived from this study.
Literature Cited
BONNER, F.T., AND T.R. DELL. 1976. The Weibull function: A new
method of comparing seed vigor. J. Seed Technol. 1:96 –103.
BRAMLETT, D.L., T.R. DELL,AND W.D. PEPPER. 1983. Genetic and
maternal influences on Virginia pine seed germination. Silvae
Genet. 32:1– 4.
BROWN, R.F., AND D.G. MAYER. 1988a. Representing cumulative
germination. 1. A critical analysis of single-value germination
indices. Ann. Bot. 61:117–125.
BROWN, R.F., AND D.G. MAYER. 1988b. Representing cumulative
germination. 2. The use of the Weibull function and other
empirically derived curves. Ann. Bot. 61:127–138.
CAMPBELL, R.F., AND F.C. SORENSEN. 1979. A new basis for
characterising germination. J. Seed Technol. 4:24 –34.
CHING, T.M. 1959. Activation of germination in Douglas-fir seed
by hydrogen peroxide. Plant Physiol. 34:557–563.
226 Forest Science 54(2) 2008
CHRISTOPOULOS, A., AND M. LEW. 2000. Beyond eyeballing: Fit-
ting models to experimental data: Critical review. Biochem.
Mol. Biol. 35:359 –391.
CZABATOR, F.J. 1962. Germination value: An index combining
speed and completeness of pine seed germination. For. Sci.
8:386 –396.
DELL, T.R., J.L. ROBERTSON,AND M.I. HAVERTY. 1983. Estima-
tion of cumulative change of state with the Weibull function.
Bull. Ecol. Soc. Am. 64:38 – 40.
EL-KASSABY, Y.A. 2000. Representation of Douglas-fir and west-
ern hemlock families in seedling crops as affected by seed
biology and nursery crop management practices. For. Genet.
7:305–315.
EL-KASSABY, Y.A., A. BENOWICZ,AND D.G.W. EDWARDS. 2002.
Genetic control of germination and aging: lessons for practice
and conservation. P. 70 –78 in Proc. of 2002 Annual Meeting of
IUFRO, Research Group for Seed Physiology and Technology:
Tree seeds 2002, Thanos, C.A., T.L. Beardmore, K.F. Connor,
and E.L. Tolentino Jr. (eds.). IUFRO, Chania, Crete.
HILL, A.V. 1910. The possible effects of the aggregation of the
molecules of haemoglobin on its dissociation curves.
J. Physiol. 40:iv–vii (as cited in Christopoulos and Lew, 2000).
HILL, A.V. 1913. The combinations of haemoglobin with oxygen
and with carbon monoxide I. Biochem. J. 7:471– 480 (as cited
in Christopoulos and Lew, 2000).
INTERNATIONAL SEED TESTING AGENCY. 2006. International rules
for seed testing 2006. Seed Sci. Technol. 21(Suppl).
KESHET, L. 2006. Calculus course notes. Dept. of Mathematics,
Univ. of British Columbia, Vancouver, BC, Canada. Available
online at www.ugrad.math.ubc.ca/coursedoc/math102/keshet.
notes/index.html; last accessed Feb. 5, 2007.
KOLOTELO, D. 1994. Response of interior spruce to extended
stratification, Vol. 7. BC Ministry of Forests, Tree Improve-
ment Branch, Victoria, BC, Canada. Seed and Seedling Exten-
sion Topics. 4 pp.
KOLOTELO, D. 2006. Quantifying seed dormancy? Can. Tree
Improve. Assoc. Tree Seed Working Group News Bull.
44:6 – 8.
KOLOTELO, D., E. VAN STEENIS,M.PETERSON,R.BENNETT,D.
TROTTER,AND J. DENNIS. 2001. Seed handling guidebook. BC
Ministry of Forests, Tree Improvement Branch, Victoria, BC,
Canada. 106 pp.
KRAKOWSKI, J., AND Y.A. EL-KASSABY. 2005. Lodgepole pine and
white spruce germination: Effects of stratification and simu-
lated aging. Silvae Genet. 54:138 –144.
LEADEM, C.L. 1986. Stratification of Abies amabilis seeds. Can. J.
For. Res. 16:755–760.
THE MATHWORKS,INC. 2005. MATLAB: The language of techni-
cal computing, version 7.1.0.246(R14) service pack 3. The
MathWorks, Inc., Natick, MA.
RICHTER, D.D., AND G.L. SWITZER. 1982. A technique for deter-
mining quantitative expressions of dormancy in seeds. Ann.
Bot. 50:459 – 463.
RINK, G., T.R. DELL,G.SWITZER,AND F.T. BONNER. 1979. Use of
the Weibull function to quantify sweetgum germination data.
Silvae Genet. 28:9 –12.
SCOTT, S.J., R.A. JONES,AND W.A. WILLIAMS. 1984. Review of
data analysis methods for seed germination. Crop. Sci.
24:1192–1199.
STOEHR, M.U., S.J. L’HIRONDELLE, W.D. BINDER,AND J.E. WEB-
BER. 1998. Parental environment aftereffects on germination,
growth and adaptive traits in selected white spruce families.
Can. J. For. Res. 28:418 – 426.
THOMSON, A.J., AND Y.A. EL-KASSABY. 1993. Interpretation of
seed-germination parameters. New For. 7:123–132.
TIPTON, J.L. 1984. Evaluation of three growth curve models for
germination data analysis. J. Am. Horticult. Soc. 109:451– 454.
Forest Science 54(2) 2008 227
