Leaf growth analysis of Ficus formosana Maxim.
I. Leaf-growth curve fitted and analysis
Chyi-Chaunn Chen and Yang-Reui Chen
Department of Botany, National Taiwan University, Taipei, Taiwan
Chyi-Chuann Chen
Address: Department of Botany, National Taiwan University, Taipei, Taiwan
E-mail : [email protected]
Keywords: Cell division . Cell elongation . Ficus formosana Maxim. . Leaf-growth
curve . Leaf Plastochron Index (LPI) . Leaf Reference Growth Days (LRGD) .
Sigmoidal curve
Abbreviations: P = Plastochron; PI = Plastochron Index; LPI = Leaf Plastochron
Index; LRGD = Leaf Reference Growth Days
Abstract
Based on the cell theory, we considered the leaf length growth were the sum
of the length of cells that arranged in the same direction with midrib. In other words,
the characteristic of the leaf length growth could be reflected by the growth conditions
of the cells during the leaf development.
In this article we proposed a simple model which was established by
analyzing the leaf length growth curve of Ficus formosana Maxim.plants with the
hope to explain the development of dicotyledonous leaves. The leaf growth data could
be best fitted by the Weibull function with two reflection points, 3.13LRGD and
8.37LRGD were obtained when differentiated the Weibull function. Furthermore,
divided the growth data in to three groups by this two reflection points and fitted this
data with exponential growth, linear, and exponential decay function, which implied
three phases: log phase, linear phase and stationary phase, respectively.
According to this model, leaf growth before the first reflection point was
mainly caused by cell division. Cell division first stopped at leaf tip region at the first
reflection point. During the first and second reflection points, the leaf was in the
process of the cell division exponentially ceased and cell elongation exponentially
increased, gradually and basipetally. Moreover, the cell division stopped at the basal
region of the leaf at the second reflection point. After the second reflection point, cell
elongation exponentially stopped. Finally, we proposed two parameters: the average
cell division rate (r), the average leaf expansion rate (V) which can be used in the
study on the effects of different environment factors on leaf growth.
Introduction
To study leaf development, the framework and analysis of the leaf growth
pattern is an important work. Several studies about the leaf growth pattern have been
reported (Erickson 1976; Charles-Edwards 1979, 1983; Lainson and Thornley 1982;
Thornley et al.1981). The spatial and temporal patterns of monocotyledon and
dicotyledonous leaves are different; therefore kinetic methods are based on different
principles (Tardieu and Granier 2000). Developing monocotyledons leaves can de
divided into cell division, elongation and maturation zones. The cell division and
tissue expansion are limited to the first centimeters beyond the leaf insertion point
(Durand et al. 1995; Schnyder and Nelson 1988). The linear phase of elongation zone
provides an easy system to analyze the responses of leaf growth to environmental
conditions (Ben Haj Salah and Tardieu 1997; Passioura and Gardner1990), and during
this phase the spatial distributions of relative elongation rate and of cell length at the
base of leaf (the growing region) are in steady state for several days (Ben Haj Salah
and Tardieu 1995; Bernstein et al. 1995; Schnyder et al. 1990). However, the
dicotyledonous leaves don’t have any distinct region to study. The measurement of
the rates of cell division and cell elongation of dicotyledonous are difficult and
complicated (Granier et al. 2000; Tardieu and Granier 2000).
In addition, the quantitative analysis and mathematic regression techniques have
been done and applied to describe the leaf development (Dennett et al. 1978; Hunt
1979, 1980; Jolliffe and Courtney 1984; Sivakumra and Shaw 1977). In recent years,
many botanists have favored the employment of fitted curves when studying the plant
growth analysis. Several mathematical functions were used to fit the leaf-growth
curve, such as the logistic growth function (Thornley 1990), the Weibell function
(Bonner and Dell 1976), a modified Weibull function (Bonner and Dell 1976), the
Richards function (Venus and Causton 1979), the Gompertz function (Berry et al.
1988), the splines function (Parsons and Hunt 1981) and polynomial function (Cottrell
et al. 1985). Moreover, often observed S-shaped leaf growth curve shows that a period
of slow growth (the lag phase) was followed by a period of growth (the logarithmic
and linear phase), which in turn was succeeded by another period in which growth
was slow or absence (the stationary phase) (Taiz and Zeiger 1991). In addition, the
cell growth contains cell division and cell elongation, and leaf development results
from cell growth with the combined effects of division, expansion, and differentiation.
It is not known whether information about the leaf cell growth pattern can be obtained
from analysis of the fitted leaf growth curve.
“Cell theory” and “organismal theory” are two theories that debated about the
plant development until today. According to the cell theory, morphogenesis in
multicellular organism results from oriented cell division and cell growth, the unit of
morphogenesis of plants is the cell (Sitte 1992); another interpretation of
morphogenesis of plants is the organismal theory, which postulates that the individual
cell is not the basic unit of morphogenesis. The cells are conceived as a subunit of
individual development (Kaplan 1992). How to explain the leaf development from
leaf growth curve in view of cell theory or organismal theory?
We proposed the working hypothesis on the basis of cell theory, the leaf length
is the sum of a layer cells growing along the long axis of leaf, and the characteristic of
the leaf-growth curve can reflect the cell growth conditions in different development
stages. How to describe the leaf development through analysis of the leaf length
growth curves is the aim of this article. We tried to know how cell division and cell
elongation to cooperate during leaf development by fitting and analysis the
leaf-growth curve of F. formosana. First, we used different sigmoidal growth
functions, such as Weibull, Gompertz, Richards, Logistic, and Hill functions to fit the
leaf length growth data and to find the best one. Second, use the fitted leaf growth
curve to identify three growth phases by two critical points obtained by mathematic
secondary differentiation. Then, data of three growth phases can be fitted by
exponential growth, linear and exponential decay functions, respectively. Their
biological meanings would be elaborated individually with three mathematic
characters of three phase of the sigmoid curve. Eventually, through those analysis
results could propose a simple growth model to explain the processes of cell division
and tissue expansion in F. formosana leaves, then this model also can be applied to
the analysis of other dicotyledonous leaves. We suggested three parameters to study
on the environment factors affecting the cell division and tissue expansion of the leaf
development.
Materials and methods
Plant material and leaf length measured
Young plants of Ficus formosana Maxim. form Shimadae Hayata were collected
from Ping-tong Country of southern Taiwan and cultured in soil pots on the bench in
green house of Botanical Department, National Taiwan University. Growth conditions
of the greenhouse were nature sunlight source, temperature and humidity were 30±2
℃, 45±5 RH in day and 25±2 ℃, 85±5 RH in night. Daily watering, recording
temperature and humidity with automatic device, recording the leaf length of
successive leaves by using a ruler at PM 5:00 every day and adding fertilizers once in
a while were held during whole experiment period. Finally, the growth data of 5-15
successive leaves on the shoot were collected for measurement and analysis.
The Plastochron Index (PI) and Leaf Plastochron Index (LPI)
The PI was calculated using the formula below (Erickson and Michelini 1957)
PI = n+(Log Ln+1 - log λ)/(Log Ln - log Ln+1) (1)
where Ln+1 was the length (Centrometers) of a leaf just shorter than λcm and Ln was
the length of the next leaf that was longer than λcm. n was the serial number of leaf.
A reference length of 1cm was found to be appropriate for this species in the article.
The PI was therefore equivalent to the distance in time between two successive leaves
reaching 1cm. Data for PI is calculated by formula (2).
PI = n+(lnLn+1 – ln 1)/(lnLn – lnLn+1) (2)
Subsequent calculation of LPI meant that a leaf exactly 1cm long would have an LPI
of 0 and the LPI was measured by formula (3)
LPI= PI – a = n-a+(lnLn+1 – ln 1)/(lnLn – lnLn+1) (3)
Where a is the serial number of the chosen leaf.
Leaf Interval Measurement Index (LIMI) and Leaf Reference Growth Day (LRGD)
The LIMI was another leaf age present method (Chen and Chen 2003).
Basically, it is an x-axis-shift transfer method about time scale and calculated using
the (4) and (5).
IMI= [lnL(i)-lnλ]/[ lnL (i)-lnL (i-1)]
(4)
LIMI = n + IMI
(5)
The time course of a single leaf, successive recorder time on a growth curve to
chose a reference leaf lengthλ. Let i denote the serial number of recorder time which
leaf length is greater than or equal to the reference length,λ. If the length of the
recorder time ith is denoted by the symbol L(i),where L(i) was the length (Centro
meters) of a leaf just larger thanλand L(i-1) was the length of the last recorder time
that was short thanλ. (i+n) was the serial number of measurement interval times in
the same leaf data.. Define the LIMI was 0, repress the leaf growth of the recorder
time to reach the reference leaf length.
When IM is 1 day, the LRGD was calculated by LIMI times 1 day. So the LRGD is
really time scale and its unit is day.
Plant leaf growth curve regressive fitting and analysis
The growth data of 5-15 successive leaves of the shoot were plotted by leaf
length to time. Transformation of the original growth data, the leaf age was expressed
in LPI or LGMI which were calculated by equation (2) to (5). Then fitted the
successive leaf growth curves using different growth type function –Sigmoidal
Growth functions were supported from the programs of the Microcal ™ Software, Inc.
ORIGIN version 6.0. Different functions formulas are listed by (6) ~ (11).
Boltzmann function ; y =A2+[A1-A2]/[1+exp[[x-x0]/dx] , dx≠0 (6)
Hill function; y =Vmax*x^n/[k^n+x^n]
(7)
SGompertz function; y =a*exp[-exp[-k*[x-xc]]] (8)
SLogistic1 function; y =a/[1+exp[-k*[x-xc]]] (9)
SRichards function; y=a*[1+[d-1]*exp[-k*[x-xc]]]^[1/[1/d]], d≠1
(10)
SWeibull 1 function; y =A*[1- exp[k*[x-xc]]^d]] (11)
Further, we compared the results of different fitted functions of the leaf growth
curve. Then analysis the best fitting function equation by second differentiation, and
to find two reflected points which can separate all leaf growth data into three groups
whether the regression curve or the original growth data. At the same time, the
exponential equation, linear function, and exponential decay equations were used to
fit three groups’ data, respectively.
Results
Plant morphology and the original growth data of the experimental leaves
F. formosanas belongs to the Moraceace family. Leaf shape is linear to
lanceolate form with 6-15 cm in long and 2-3 cm in wide. Phyllotaxis is spiral that
leaf primordium spirally arrange on the shoot, as shown on Fig. 1.
The original data of the F. formosana leaves were presented in the Fig. 2A-B.
We measured and recorded the first 5 to 15 successive leaves on the first branch shoot.
Each growth curve shows a sigmoid curve (Fig. 2A). The linear relationship of the
logarithmic leaf length plotted against time in early stage indicating that it is an
exponential growth (Fig. 2B).
Calculation of the leaf age – LPI and LIMI
All leaf growth data transformed and let the leaf age expressed in LPI or LIMI.
LPI and LIMI were calculated according by the Erickson and Michelini (1957)
methods and the x-axis-shift transfer method (Chen and Chen 2003), respectively.
According the result of Fig. 2B, selected 1cm as the reference length λ and
transformed data. Transformed data their time expressed by LIMI were shown in Fig.
3A,C and presented by LPI were shown in Fig. 3B,D. The leaf growth curve can fit
with sigmoidal function (Fig. 3A,B). In a same time, the LN (leaf-length) plotted
against time also can fit with cubic polynomial function (Fig. 3C,D). To compare Fig.
3A(C) with Fig. 3B(D) the variability of the data presented in LPI method that were
higher than in LIMI method. LPI and LIMI present a positive relationship R=0.99; the
linear relationship was LPI=0.0264+0.5874*LIMI (Fig. 4). However, it seems
observed that leaf age expressed with LIMI is better than LPI. According to the
definition of LIMI, when IM is 1day the leaf age can represent with LRGD and its
unit is day. So for the subsequent data, time scale was expressed in LRGD.
Leaf transformed data were fitted by several growth type functions
In order to get best fitting function of the leaf growth curves, we selected and
compared with several sigmoidal growth type functions such as Boltzmann, Hill,
SGompertz, SLogistic, SRichard, and SWeibull functions to fit (Fig. 5A-F). All fitted
results were shown in Table 1. It is clear that the best-fitted serious orders are
SWeibull, SRichard2, Boltzmann, SLogistic1, SGompertz, and Hill equations.
Analysis of the leaf growth curve by differentiated
As shown in the table 1, the best- fitted function was the SWeibull function,
f(x)=11.44941*(1-exp(-0.01562*(x+58.05158))^23.90726), and f(x) equation was
differentiated by first differentiation and secondary differentiation, as shown in Fig.
6A-B. The relative growth rate reaches the maxims 1.57cmday-1 at 5.57 LRGD (Fig.
6A), the secondary differentiation graph can obtain two reflective points, LRGD are
3.13 and 8.37, respectively (Fig. 6B).
Fit the theoretical leaf growth curve and original leaf growth data by mathematical
functions
In order to estimate the correction of the analysis, we further used two reflective
points to separate the theoretical leaf growth curve data and original leaf growth data
into the a, b, c and a’, b’, c’ groups, respectively (data not shown). Then those data
can be fitted by suitable mathematical functions. All groups’ data can obtain the
best-fitted functions (Fig. 7A-B) and all equations were shown in Table 2. The results
showed that not only the theoretical leaf growth curve data but also original leaf
growth data would be fitted with exponential growth, linear polynomial and
exponential decreased functions, respectively. In the theoretical leaf growth curve data,
the two points almost could fully identify the three phases there have the exponential
growth, linear growth and exponential decreased growth characteristics (Ra2= 0.99;
Rb2=0.99; Rc2=0.99) (Table 2, left column). Moreover, similar to the theoretical
growth data, the original growth data could be fitted used exponential growth, linear
polynomial and exponential decreased functions, respectively (Ra’2= 0.99; Rb’2=0.99;
Rc’2=0.44)(Table 2, right column). However, both theoretical growth curve data and
actual growth curve data can classified into three growth phases, and be fitted with
optimum functions.
Discussion
The growth model for dicotyledonous leaves
Based on the present analysis, several results can be observed. 1) Sigmoid leaf
growth curve contain the log, the linear, and stationary phases, which can be separated
by two inflective points, obtained from secondary differentiation of the best-fitted
function. 2) Three growth phases can be best fitted by exponential growth, linear, and
exponential decay functions, respectively. 3) The exponential growth function implied
that the leaf cells are at the cell division stage on the log phase. At the same time, the
linear phase infers that cells are at elongation stage. 4) The exponential decay is
means that the leaf elongation cease is related with the process of cell division stop.
We proposed a simple growth model to describe the leaf development of
dicotyledonous leaves showing in the Fig.8A-B. The leaf development in early stage
(log phase) is under cell division at entire leaf. The secondary phase (linear phase) of
leaf growth under temporal changes, that cell division first stopped at leaf tip then
cells numbers change basipetally and gradiently, which are going in division
exponentially decay and in elongation exponentially increased. The third phase
(stationary phase), the cell numbers of elongation was stopping in exponential
decreased. As shown on the Fig. 8A, the leaf length sigmoid growth curve is the result
of the function of cell division and cell elongation. The cell behavior contains cell
division and cell elongation temporally expressed in the process of leaf development,
as shown in Fig. 8B. How to explain the spatial expression of cell behavior in leaf
growth by this model? We suggested that leaf growing before the first reflection point
was the cell division, and at the first reflection point that cell division first stopped on
the leaf tip region. The leaf development between the first and second reflection
points was in the process of the cell number change in division ceased and elongation
increased exponentially and basipetally. At the second reflection point was the cell
division stopped lastly on the basal region of the leaf. Follow the second reflection
point implied the cell elongation exponentially stopped in the leaf development.
This model confirms the result of sunflower (Helianthus annuus L.) leaf
development proposed by Granier and Tardieu (1998a, b). That leaf development
consists of a three-phase process with the transitions occurring with a tip-to-base
gradient within the leaf. During the first period increases in area and cell number in
the leaf zone are both exponential. A second period follows with a decline in RDR but
with a maintained RER. During the third period RER declines. This three-phase
development is observed in each leaf zone, but periods with exponential expansion
and with exponential division are shorter near the leaf tip than near the leaf base.
The analysis of Fleming (2002) study indicates the existence of both cell
division-dependent and cell division-independent mechanisms in leaf morphogenesis
and highlights the importance of future investigations to unravel the co-ordination of
these mechanisms. The mechanism of cell division-independent morphogenesis was
the cell wall and growth rate and the mechanism of cell division-dependent
morphogenesiswas cell division and growth rate. They suggested that leaf
development was through the integration and control of division-dependent and
division-independent mechanisms of morphogenesis. Our model can explain the
explanation of two mechanisms, the mechanism of cell division-independent
morphogenesis affected through leaf expansion rate and the cell elongation rate and
the mechanism of cell division-dependent morphogenesiswas affected through the cell
division rate. Nonetheless, the mechanisms controlling these basic aspects of leaf
development remain to be characterized since the pattern of growth within the leaf
blade is quite complex (Poethig and Sussex 1985; Steeves and Sussex 1989).
Cell division is not the only mechanism that determines final leaf area (Green 1976),
because cell division and tissue expansion are closely coordinated during leaf
development (Jacobs 1997; Neufeld and Edgar 1998; Granier et al. 2000). We think
close coordination of two events is based on the cell number of cell division during
leaf development (Fig. 8B) In principle; the number and size of leaf cells affect the
dimension and size of the leaf. However, leaf size is partially uncoupled from cell size
and number by a compensatory system(s). An understanding of this compensatory
system(s) at the molecular and genetic levels will enhance our understanding of the
mechanism of leaf morphogenesis, and subsequently, of the mechanisms that control
morphogenesis in multicellular organisms. Moreover, progress in understanding
cell-cell communication (particularly between the L1 and other layers) and regulation
at the whole-plant level will improve the knowledge of the control of leaf shape and
size. (Tsukaya 2003)
Application of the growth model
How to apply the growth model? There were two parameters in the analysis of
the model that are the average growth rate in linear phase and average cell division
rate in the log phase. How to obtain the average division rate (r) in the log phase, we
can calculate by formula (12) and (13)
(Ln2/Lw)=(Ln1/Lw)*2 (t2-t1)r (12)
r = log (Ln2/Ln1)/t* log2
(13)
Ln1 and Ln2 are the leaf length at t1 and t2 in the log phase; r is the average cell
division rate.
The average leaf elongation rate (V) in leaf expansion can calculate by formula (14)
V = (Ln4-Ln3)/(t4-t3) (14)
Ln3 and Ln4 are the leaf length at t3 and t4 in linear phase; V is the average leaf
expansion rate.
Estimate of environmental influences on leaf expansion
The variables of the aerial environment that may potentially influence leaf
expansion include temperature, light and carbon dioxide. Soil variables that influence
leaf expansion including water and mineral nutrient availability, the salt concentration
of the soil solution and soil temperature (Terry et al. 1983). This model can propose
two parameters to indicate the effects of different environment factors. In other words,
through measured and analysis the leaf growth curves by this model and used formula
(12) ~ (14) to calculate the average cell division rate (r), the average leaf expansion
rate (V). The study of relationship between these parameters and environmental
factors can understand the factors affecting the leaf development.
The cell growth patterns resulted from the combined effects of division,
expansion, and differentiation during leaf development. Leaf organogenesis is
explained from the perspective of the cell theory. According to this theory, the cell is
the basic unit of multicellular organism; therefore, the unit of organogenesis or
morphogenesis should be the cell. The leaf growth curve is the result of quantitative
growing of cells which containing the cell division and cell elongation. However, this
model proposed and explained that leaf-length growth curve is aggregation of a layer
cells under cell division and cell elongation in the process of the leaf growth, but
cannot explain cell differentiations in the inner tissue layers of leaf development.
Therefore, we suggested further section the leaf to obtain the growth information
about different tissues during leaf development to understand the relationship between
the different tissue layers.
Fig. 1 Morphology of the Ficus formosana Maxim. Plant. Leaf shape is
linear-lanceolate, 6-15 cm long 2-3 wide, and a spiral phyllotaxis.
14
leaf 5
leaf 6
leaf 7
leaf 8
leaf 9
leaf 10
leaf 11
leaf 12
leaf 13
leaf 14
leaf 15
Leaf length (cm)
12
10
8
6
4
2
A
0
0
2
2.8
6
8
10
12
14
16
18
20
8
10
12
14
16
18
20
leaf 5
leaf 6
leaf 7
leaf 8
leaf 9
leaf 10
leaf 11
leaf 12
leaf 13
leaf 14
leaf 15
2.4
LN( leaf length, cm )
4
2.0
1.6
1.2
0.8
0.4
0.0
-0.4
B
-0.8
0
2
4
6
Time (days)
Fig. 2A-B Time course of leaf expansion for 5-15 successive leaves from the shoot of
Ficus formosana Maxim. plant. A The original growth data, which show leaf length
plotted against with time. B The original growth data, which show LN (leaf length,
cm) plotted against with time.
14
14
leaf length vs LIMI
Sigmoid regression
10
8
6
4
10
8
6
4
A
2
leaf length vs LPI
Sigmoid regression
12
Leaf length (cm)
Leaf length (cm)
12
B
2
0
0
-4
-2
0
2
4
6
8
10
12
14
-3
16
-2
-1
0
1
2
2.8
4
5
6
7
8
9
10
5
6
7
8
9
10
2.8
LN(leaf length) vs LIMI
Cubic regression
2.4
LN(leaf length) vs LPI
Cubic regression
2.4
2.0
LN(leaf length, cm)
2.0
LN(leaf length, cm)
3
LPI
LIMI
1.6
1.2
0.8
0.4
0.0
-0.4
1.6
1.2
0.8
0.4
0.0
-0.4
C
-0.8
D
-0.8
-4
-2
0
2
4
6
8
10
12
14
16
-3
LIMI
-2
-1
0
1
2
3
4
LPI
Fig. 3A-D. The leaf growth curves of leaf growth data of Ficus formosana Maxim.
plant. Leaf age of the growth data expressed in LIMI or LPI. A,B Leaf length plotted
against time. The growth data can be fitted with sigmoidal function.
A y = 11.943/(1+ exp (-(x-5.015)/1.9829)); R2 =0.996.
B y = 11.9072/(1+ exp (-(x-2.9538)/1.2146)); R2 =0.979.
C,D LN (leaf length) plotted against time. Fit the data with cubic function.
C y =0.0402+0.4441x-0.0227x^2+0.0002x^3; R2 =0.995.
D y =0.0355+0.774x-0.0726x^2+0.0017x^3; R2 =0.987.
10
9
LPI vs LIMI
LPI = 0.0264 + 0.5874 LIMI
8
7
6
LPI
5
4
3
2
1
0
-1
-2
-2
0
2
4
6
8
10
12
14
16
LIMI
Fig. 4 The relationship of LPI and LIMI. LPI and LIMI is positive correlation, R=0.99;
IM=1.
14
14
A Boltzmann
12
10
Leaf length (cm)
10
Leaf length (cm)
B Hill
12
8
6
4
8
6
4
2
2
0
0
-4
-2
0
2
4
6
8
10
12
14
16
-4
14
2
4
6
8
10
12
14
16
2
4
6
8
10
12
14
16
D SLogistic1
12
12
10
10
Leaf length (cm)
Leaf length (cm)
0
14
C SGompertz
8
6
4
8
6
4
2
2
0
0
-4
-2
0
2
4
6
8
10
12
14
16
-4
14
-2
0
14
F SWeibull 1
E SRichard2
12
12
10
10
Leaf length (cm)
Leaf length (cm)
-2
8
6
4
8
6
4
2
2
0
0
-4
-2
0
2
4
6
LRGD
8
10
12
14
16
-4
-2
0
2
4
6
8
10
12
14
16
LRGD
Fig. 5A-F The regression of the leaf-growth curves of F. formosana Maxim fitted by
different growth type functions (Time is expressed with LRGD). A Boltzmann
equation. B Hill equation. C SGompertz equation. D SLogistic1 equation. E
SRichard2 equation. F SWeibull equation.
12
A
_f(x)
Leaf length (cm)
10
8
6
4
2
0
-4
-2
0
2
4
6
8
10
12
14
16
--- f(x)'
__ f(x)"
B
0.4
d(Leaf length, cm)/dt ---
1.5
0.2
1.0
0.0
-0.2
0.5
-0.4
0.0
d(Leaf length, cm)/dt*1/dt __
0.6
2.0
-0.6
-4
-2
0
2
4
6
8
10
12
14
16
LRGD
Fig. 6A-B Analysis of the fitted leaf-growth curve of Ficus formosana Maxim. by
first and secondary differentiations. A The best fitted leaf-growth curve regression
with the Weibull function f(x) , y =11.44941*(1-exp(-0.01562*(x+58.05158))^
23.90726) . B Graph of the first and secondary differentiation. Dash line indicate the
graph of first differentiation f(x)’, d(leaf length, cm)/dt plotted against with time, the
maximum is 1.57 cm day-1 when LRGD is 5.57. Solid line indicate the graph of the
secondary differentiation f(x)”, d(leaf-length)/dt*1/dt plotted against time, the
maximum and minimum are 0.26 cm*day-1 (LRGD=3.13) and –0.46 cm*day-1
(LRGD=8.37) , respectively .
14
A
12
Leaf length (cm)
10
8
6
a group
b group
c group
f=y0+a*exp(b*x)
f=y0+a*x
f=y0+a*exp(-b*x)
4
2
0
-4
-2
0
2
4
6
8
10
12
14
16
18
14
B
12
Leaf length (cm)
10
8
6
a' group
b' group
c' group
f=y0+a*exp(b*x)
f=y0+a*x
f=y0+a*exp(-b*x)
4
2
0
-4
-2
0
2
4
6
8
10
12
14
16
18
LRGD
Fig. 7A-B Fitting of leaf-growth data of Ficus formosana Maxim that separated with
two reflected points obtained by secondary differentiation the leaf-growth curve.
Three groups of growth data were fitted with exponential growth (___) , linear (---),
and exponential decreased equations (…), respectively. A The fitting curve data. B
The original growth data. The a, b, and c groups data were separated by LRGD are
3.13 and 8.37.
Fig. 8A,B The analysis model to describes the leaf development pattern of the Ficus
formosana Maxim. plant. A Leaf growth curve contain three phases: log phase, linear
phase, and stationary phase. B Partition of cell numbers in cell division or cell
elongation in the leaf development.
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