# 2 Germination variables

## 2.1 Germinability (G)

According to L. G. Labouriau (1983a), germinability is the percentage of seeds in which the germination process comes to the end in the experimental conditions by the intraseminal growth, resulting in the protrusion (or emergence) of a living embryo.

$g=\left(\frac{\sum_{i=1}^kn_1}{N}\right)100 \\ (in\ percentage;\ expressed\ in\ the\ program\ as\ GRP)$

Where: $$n_i$$: number of germinated seed in the $$i^{th}$$ time; N: total number of seed in each experimental unit.

## 2.2 Mean Germination Time (t) and Standard Deviation (St)

The mean germination time was proposed by Haberlandt in 1875 (L. G. Labouriau 1983a) and used by Czabator (1962) as mean length of incubation time (see Ranal and Santana (2006) for other details). It is calculated as the weighted mean of the germination time (hour, day or other time unit). The number of germinated seeds at the intervals established for the data collection is used as weight (Ranal and Santana 2006).

$\overline{t}=\frac{\sum_{i=1}^kn_it_i}{\sum_{i=1}^kn_i} \\ (in\ hour,\ day\ or\ other\ time\ unit; \ expressed\ in\ the\ program\ as\ MGT)$

$S_t=\sqrt{\left\{\frac{\left\{\sum_{_{i=1}}^kn_i\left(t_i-\overline{t}\right)^2\right\}}{\sum_{_{i=1}}^kn_i-1}\right\}} \\ (in\ the\ same\ time\ unit\ of\ the\ mean\ germination\ time;\ expressed\ in\ the\ program\ as\ SDG)$

Where $$\bar{t}_i$$: time from the start of the experiment to the $$i^{th}$$ observation (hour, day or other time unit); $$n_i$$: number of germinated seeds in the $$i^{th}$$ time (not the accumulated number) and $$k$$: the last time of observation.

## 2.3 Coefficient of Variation of the Germination Time

$CV_t=\frac{\left\{S_t\right\}}{\overline{t}}100 \\ \left(in\ the\ same\ tine\ unit\ of\ the\ mean\ ger\min ation\ time;\ \exp ressed\ in\ the\ program\ as\ SDG\right)$ where $$S_t$$: standard deviation of the germination time and $$\overline{t}$$: mean germination time.

## 2.4 Mean Germination Rate (v)

The mean germination rate is defined as the reciprocal of the mean germination time, since the mean germination rate increases and decreases with $$\frac{1}{\overline{t}}$$, not with $$\overline{t}$$ (L. G. Labouriau 1983b).

$\overline{v}=\frac{1}{\overline{t}} \\ \left(in\ hours^{-1},\ day^{-1}\ or\ other\ reciprocal\ time\ unit,\ \exp ressed\ in\ the\ program\ as\ MGR\right)$

## 2.5 Uncertainty Index (U)

This measurement is an adaptation of the Shannon index and measures the degree of uncertainty associated to the distribution of the relative frequency of germination (Labouriau and Valadares 1976). Low values indicate more synchronized germination (Ranal and Santana 2006).

$U\ =\ -\sum_{i=1}^kf_i\log_2f_i\ \\ \left(in\ bit;\ \exp ressed\ in\ the\ program\ as\ UNC\right),\ being\ f_i=\frac{n_i}{\sum_{i=1}^kn_i}$

Where $$f_i$$: relative frequency of germination; $$n_i$$: number of seed germinated in the $$i^{th}$$ time, and $$k$$: the last day of germination.

## 2.6 Synchronyzation Index (Z)

This index was proposed by Primack (1985) to assess the degree of overlapping of flowering among individuals in a population and Ranal and Santana (2006) adopted it for seed germination. The synchrony of germination of one seed with another assumed $$Z=1$$ when the germination of all seeds occur at the same time and $$Z = 0$$ when at least two seeds can germinate, one at each time.

$Z=\frac{\sum_{ }^{ }C_{n_1,2}}{N} \\ \left(a\dim ensional\ measurement;\ expressed\ in\ the\ program\ as\ SYN\right); \\ being\ C_{n_1,2}=\frac{n_i\left(n_i-1\right)}{2}\ ;\ N=\frac{\sum_{ }^{ }n_i\left(\sum_{ }^{ }n_i-1\right)}{2}$

Where $$C_{n_i,2}$$: combination of germinated seeds in $$i^{th}$$ time, two together, and $$n_i$$: number of germinated seed in the $$i^{th}$$.

## 2.7 Limits and units of the germination variables in GerminaQuant

Table 2.1: Germination variables evaluated in GerminaQuant and limits according Ranal and Santana (2006) ; where: $$n_i$$, number of seed germinated in $$i^{nth}$$ time ; $$K$$, the last day of the avaliation process for germination.
variables abbreviation limits units
germinated seed number GSN $$0 \le n \le n_i$$ $$count$$
germinability GNP $$0 \le g \le 100$$ $$\%$$
germination asin ASG $$0 \le arsin \le 1$$ $$grade$$
mean germination time MGT $$0 \le t \le k$$ $$time$$
germination speed SPG $$0 < g_s \le 100$$ $$\%$$
mean germination rate MGR $$0 < v \le 1$$ $$time^{-1}$$
Synchronyzation Index SYN $$0 \le Z \le 1$$ $$-$$
Uncertainty Index UNC $$0 \le U \le log_2 n_i$$ $$bit$$
germination standard deviation SDG $$0 < s_t^2 \le \infty$$ $$time^2$$
germination variance VGT $$0 < s_t \le \infty$$ $$time$$
Coefficient of variation CVG $$0 < CV_t \le \infty$$ $$\%$$