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Published in Agron. J. 97:41-48 (2005).
© American Society of Agronomy
677 S. Segoe Rd., Madison, WI 53711 USA

Agronomic Modeling

A Mechanistic Model for Describing Corn Plant Leaf Area Distribution

Jinzhong Yanga and Mark Alleyb,*

a Dep. of Agron., Shanxi Agric. Univ., Shanxi, China
b Dep. of Crop and Soil Environ. Sci., Virginia Tech, Blacksburg, VA 24061-0403

* Corresponding author (malley{at}vt.edu)

Received for publication September 12, 2003.

   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Mathematical models to describe corn (Zea mays L.) leaf area are important components of computer simulation of corn growth and development. The objective of this research was to develop a mechanistic model to describe corn leaf growth. The hypothesis was that single leaf growth rate is the difference between potential positive leaf growth and reduced leaf growth rate because assimilates are also being utilized for stem, roots, other leaves, and reproductive organ growth. The model is S = S0exp[(L – L0)2/(–2k2)], where S is the area of individual leaf with rank L, S0 is the area of the largest individual leaf with rank L0, and k is a constant. The model was evaluated with 77 independent data sets from 64 genotype-by-environment combinations during 1989 to 2001. The fitting precision of the model to the data was high (multiple correlation coefficients ≥ 0.97), the model was applicable to widely different combinations of genotypes and environments, and the model was suitable for all leaves and green leaves only. Validation of the model for predicting leaf area was conducted using 174 data sets from 30 diverse cultivars and showed that the predicted leaf area was within 10% of the measured leaf area for 27 of the 30 cultivars with maximum deviation being 14%. The correlation of predicted leaf area with measured leaf area equaled 0.94 for all data. The reduced model for individual leaf area estimation is F = F0exp{–[F0(L – L0)]2/0.3542}, where F = S/ST and ST is sum of all leaf areas.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
CORN LEAF AREA development models for both whole plants and individual leaves have been proposed by various researchers because of the importance of leaf area to photosynthesis and yield (Montgomery, 1911; Ledent, 1978; Li, 1996; Birch et al., 1998; Dwyer and Stewart, 1986; Stewart, 1993; Yang and Lu, 1992, 1996; Yang et al., 1998; Guo et al., 1999). These models have described leaf area changes as plant populations change, and leaf area of individual plants and individual leaves, and form the basis for computer simulation of corn growth and development.

Dwyer and Stewart (1986) introduced a four-parameter, slightly skewed modified bell-shaped model to describe the relationship between leaf rank and the area of individual mature leaves. For the relation between individual leaf area and leaf rank on the corn plant, Yang et al. (1998) developed a four-parameter modified S-shaped model. Parameters in both these models do not have explicit biological meaning but are leaf area and development observations associated with temperatures and available water measurements. Keating and Wafula (1992) showed that the model parameters developed by Dwyer and Stewart (1986) could be related geometrically. These model parameters were also shown to be related to total leaf number in certain situations (Birch et al., 1998). In addition, Elings (2000) observed that it is difficult to predict these parameters in advance and that the predictive use of the models is limited.

A somewhat different approach to the modeling of corn leaf area showed that the total plant leaf area could be estimated from the area fraction of the largest leaf and its area (Elings, 2000; Yang et al., 2000). This observed relationship occurs for two reasons. First, corn plants with different total leaf area have identical individual leaf area distribution, i.e., the largest leaf locates at a constant rank on the plant, and the area fraction of leaf with same rank is constant within the same total number of leaves (Yang and Lu, 1992). Second, "the area of the largest leaf relative to total plant leaf area is constant, and this constant is linearly related to total leaf number" (Elings, 2000). The objectives of this paper are (i) to report the development of a mechanistic model that represents individual corn leaf area distribution in relation to leaf rank on the corn plant; and (ii) to determine if the derived model accurately describes individual leaf area distribution of all leaves on the whole plant and of green leaves at anthesis for diverse corn germplasm and growing conditions.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Model Development
Corn plant development can be divided into vegetative and reproductive development stages. Wang (1999) reviewed the primary growth pattern of a corn plant and indicated that it's growth can be described by (i) dominant leaf growth, (ii) dominant stem growth, and (iii) dominant reproductive organ growth and that these three growth phases overlap. The rank of the most recently mature leaf can be used as an indicator for timing organogenesis of root, stem, tassel, and ear, albeit the underlying genetic mechanism is still unclear (Wang, 1999). Thus, we may use leaf rank as a measure of biological time as opposed to using physical time, i.e., days after emergence, when analyzing plant growth and development. Finally, we may assume that leaves are the only source of assimilates for plant growth. However, the assimilate source for leaf growth originates from previously developed leaves, but assimilate available for leaf growth declines as leaf rank increases because assimilates are utilized for sink organs such as stem, roots, ear, and tassel growth. Furthermore, the assimilate needed by sink organs increases as leaf rank increases because the plant roots, stem, and ear are increasing in size. For example, kernel development requires the stored or structured form of assimilates in stems (Wang, 1999). This growth of sink organs is associated not only with assimilates directly from leaves, but it is also associated with leaf rank, which may represent some genetic trigger. In addition, sink organ growth exerts a negative influence on the growth of leaves, and the growth rate of a new leaf can be viewed as the difference between positive growth factors (total assimilate available for total plant growth) and negative leaf growth factors (assimilate utilized for stem, root, and reproductive organ growth). Modeling leaf area increase as a function of leaf rank assumes that the positive growth rate of a new leaf is proportional to the size of the previously developed leaf on the plant and that the negative effect on growth rate is proportional to both the size of the previous leaf and its rank. This relationship can be described as

[1]
where S is area (cm2) of the leaf with rank L; L = 1, 2, ..., the total number of leaves (n); and both a and b are positive constants. Equation [1] indicates that the negative effect on growth rate is regulated by b, which is associated with the partition of photosynthate to reproductive organs and the stem and roots. The purpose of growth, from an evolutionary view, is propagation (to reproduce kernels in corn). Thus, both stem and root growth can be considered to be subordinate to reproductive organ growth in terms of competition for assimilates. As a result, the larger the b value, the greater the amount of assimilate that may go to reproductive organs. Equation [1] is a differential equation, and its universal solution takes the form

[2]
where c is a positive constant.

Corn leaf area distribution is a curve with a single peak (Yang and Lu, 1992), and thus, individual leaf area, S0, has a maximum value at leaf rank L0. Based on this initial condition, a special solution is found for Eq. [2]

[3]

If k = b–0.5, Eq. [3] can be simplified as

[4]
where S denotes individual leaf area, L denotes leaf rank, and S0 is the leaf area of the largest individual leaf with rank L0. The equation has a maximum point of (L0, S0) and two inflection points of . Equation [4] can be used to describe individual leaf area distribution on a corn plant, and may be used to describe the relationship between leaf area and leaf rank, while Eq. [1] is its differential form.

Leaf area distribution of corn plants with different total leaf area may be compared if individual corn plant leaf area is transformed into leaf area fraction: F = S/ST, where F is leaf area fraction, S is individual leaf area, and ST is total leaf area of the plant. Thus, Eq. [4] becomes

[5]
where F0 is the largest leaf area fraction for leaf L0 with the largest leaf area S0, i.e., F0 = S0/ST.

Observations for Model Parameterization
Data for the model parameterization process were collected from a range of field experiments conducted from 1989 through 2001 (Table 1). Maximum length and width of mature leaves were measured on different, or fixed if applicable, sampling plants at varied time intervals from the full expansion of the first leaf to the silking stage or once at silking. All measurements were made only one time for a single leaf after its leaf collar was visible because previous research has shown that the leaf area does not vary after the leaf collar is visible (Hu, 1986). Leaf area was estimated using the formula: area = 0.75 x maximum leaf length x maximum leaf width, which was derived by Montgomery and used by other authors, e.g., Hu (1986), Birch et al. (1998), and Elings (2000). Averages from several plants, when applicable, or raw data from a single plant were used to establish and validate model parameters. A data set consisted of all values from leaves of corn plants with the same number of leaves under the same growing conditions. In addition, some data sets were utilized from the publications of Dwyer and Stewart (1986), Bollero et al. (1996), and Whigham and Woolley (1974). Thus, we utilized a total of 77 sets of data from 64 different combinations of genotypes and growth conditions, of which 50 sets were from all mature leaves developed on plants since emergence; the other sets were from green leaves at the silking stage. Note that the number of data sets (77) is larger than the number of genotype-by-growth condition combinations (64). This difference occurs because each data set consists of plants with the same number of leaves, and thus plants from one genotype-by-growing condition combination have been grouped into different data sets according to their number of leaves. Data sets were each separately fitted to Eq. [5].


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  		Table 1. Data source descriptions of data for the model evaluation: genotypes, environments, year, site, and the number of plants measured (PN).{dagger}

 
Observations for Model Validation
Data for the model validation were collected from a field experiment conducted under irrigation in Shanxi, China, in 2001, with 30 cultivars. The cultivars were 3119, An Yu 5, Ben Yu Jiu, Deng Hai 1, Deng Hai 3, Dun Yu 1, Hu Dan 4, Ji Dan 17, Lu Dan 50, Lu Yu 10, Lu Yu 16, Nong Da 108, Nong Da 3138, Nong Da 321, Shan Dan 902, Shan Nong 3, Shan Zi 1, Sheng Dan 7, Si Mi 25, Tang Kang 5, Tang Yu 10, Xi Nong 911, Ye Dan 13, Ye Dan 2, Yu Yu 22, Yu Yu 25, Zheng Dan 14, Zhong Dan 2996, Zhong Dan 321, and Zhong Yuan Dan 32. Four to seven plants from each of the cultivars were tagged after emergence. Mature leaf area was determined as described in the previous section. There were a total of 174 independent data sets, each from a single plant. Total plant leaf area from each of these data sets was used as the observation for model validation.

Statistical Methods
Equations [4] and [5] have the same form as the Gaussian equation, and thus it was possible to utilize CurveExpert (Hyams, 1997) to estimate parameters in the model from each set of data collected. CurveExpert is curve-fitting software that calculates the model parameters L0, F0, k, multiple correlation coefficient (R), and the standard error (SE) for a given data set. Individual leaf areas were transformed into fractions of the whole-plant leaf area (relative leaf area) to facilitate comparisons among different models. Leaf rank is counted from the top down for all data sets to facilitate the model fitting to data of green leaves at silking. Note that counting for leaf rank from bottom to top and that from top to bottom are mathematically transformable, that is, leaf rank from the top down equals the total number of leaves – leaf rank from bottom to top + 1. Leaf rank counting methods for Eq. [4] and [5] do not alter leaf area distribution on corn plants. Summary statistics such as maximum, minimum, and mean were calculated with MS Excel software for the estimates of L0, F0, k, R, and SE for all data sets. Similar summary statistics were calculated for total plant leaf area and total leaf number from the observation data.

Relations among parameters from the observations utilized for model parameterization were determined with regression analysis techniques, and the relations were used for model validation on independent data sets. Model performance was evaluated by estimate error, which was defined as mean of the absolute difference between the observed and predicted total plant leaf area, that is


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Fitting the Models Using Data for All Plant Leaves
Corn plant leaf area distributions were well fitted with Eq. [5] (Table 2 and Fig. 1). Multiple correlation coefficients for the model fitting ranged from 0.9755 to 0.9987 over a range of genotypes with total leaf area varying from 2921 to 8703 cm2 on whole plants, and whose total leaf number varied from 14 to 23 leaves. These data support the use of Eq. [5] to describe the leaf area distribution of all leaves on corn plants and indicate that a symmetric curve with three parameters adequately describes leaf area distribution. Equation [5] has fewer parameters than the skewed bell-shaped curve with four parameters presented by Dwyer and Stewart (1986), which has been used by others, e.g., Keating and Wafula (1992) and Elings (2000). Equation [5] takes the same form as Eq. [4], except for the transformation of S0 into F0 using F0 = S0/ST. For a given set of data, ST is a constant, and thus values of k, L0, multiple correlation coefficient, and other statistics from fitting Eq. [5] remain unchanged when fitting Eq. [4] while S0 = ST x F0. Equation [4] is supported wherever Eq. [5] is supported.


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  		Table 2. Summary statistics of parameter estimates, multiple correlation coefficient (R), and standard error (SE) from the model fitting, plus summary statistics of observed plant leaf areas (TLA), grouped by the number of leaves.{dagger}

 


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  		Fig. 1. Calculated leaf area fractions (F) as a function of downward leaf rank (L) for eight data sets. Each data set represents eight different corn cultivars. Cultivar names are found in the figures. The fitted model is F = F0exp[–0.5(L – L0)2/k2] (Eq. [5]), where F = individual leaf area/plant leaf area, L0 is the downward leaf rank with the maximum area fraction of F0, and k is a constant. R is the multiple correlation coefficient from model fitting.

 
Predicted maximum leaf area fractions ranged from 0.0792 to 0.1580 and occurred from Leaf 4 to 8, top-down ranking (Table 2). These were all ear leaves or the leaf next to the ear leaf. However, the coefficient k estimates exhibited large variation as values ranged from 2.58 to 5.35, and variation was large even for plants with the same total number of leaves. For example, for a plant with 20 leaves, the coefficient k values varied from 3.48 to 5.14.

The magnitude of the k coefficient may relate closely to the potential size of the corn kernel sink for photosynthate at anthesis in the sense that k is a decreasing function of b, and b is a factor determining the partition of photosynthate to reproductive organs and stem and root (see the rationale of the presented model and the premise of Eq. [1]). The smaller the k value, the greater the amount of assimilate that may go to reproductive organs. This relationship indicates that k values may be a useful selection tool in corn-breeding programs that desire to improve the amount of photosynthate going into grain. However, the data needed to determine k may preclude corn breeders with high labor costs from using this relationship. Further research efforts are needed to verify the relationships between corn vegetative and reproductive growth in terms of F0 and k values.

Fitting Models Using Data for Green Leaves at the Silking Stage
Green leaf area distributions at silking were well fitted to Eq. [5] with multiple correlation coefficients ranging from 0.9711 to 0.9979 (Table 3 and Fig. 2). Eight corn genotypes were tested with total green leaf area at silking measurements ranging from 2803 to 8983 cm2 per plant, and the number of green leaves at silking varied from 8 to 15. Also, the predicted maximum leaf area fraction, F0, values varied from 0.0943 to 0.2026, and the k coefficient values ranged from 2.35 to 4.95 (Table 3 and Fig. 2). These data support the use of Eq. [5] to describe the leaf area distribution of green leaves on corn plants at silking.


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  		Table 3. Summary statistics of parameter estimates, multiple correlation coefficient (R), and standard error (SE) from the model fitting for green leaves on the plant at silking stage, plus summary statistics of observed plant leaf areas (TLA) and of the number of all green leaves (GLN).{dagger}

 


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  		Fig. 2. Calculated leaf area fractions (F) as a function of downward leaf rank (L) for green leaves at silking stage. Each data set represents eight corn subspecies by physical and chemical traits of kernels, namely, Z. mays L. subsp. indurata Sturt, Z. mays L. subsp. indentata Sturt, Z. mays L. subsp. amylacea Sturt, Z. mays L. subsp. saccharata Sturt, Supersweet Mutant, Z. mays L. subsp. ceratina Kulesh, Z. mays L. subsp. everta, and Z. mays L. subsp. tunicata Sturt. The fitted model is F = F0exp[–0.5(L – L0)2/k2] (Eq. [5]), where F = individual green leaf area/plant green leaf area, L0 is the downward leaf rank with the maximum area fraction of F0, and k is a constant. R is the multiple correlation coefficient from model fitting.

 
Model Parameter Relations for All Leaves on Corn Plants
Standardizing the total sum of all leaf fractions on the plant equal to 1.0, correlation of the maximum leaf area fractions with the total number of all leaves on the plant was highly negative (r = –0.9651, N = 50, P < 0.001), and the linear regression equation describing the relationship was

[6]
where F0 is the same as in Eq. [5], and n is the total number of leaves.

Correlation of the rank of the leaf with the maximum leaf area to the total number of all leaves on the plant was highly positive (r = 0.9860, N = 50, P < 0.001). The linear regression equation describing this relationship was

[7]
where L0 is the same as in Eq. [5] and n is the total number of leaves.

Elings (2000) reported similar equations relating F0 and L0 to total leaf number, albeit with different coefficients. Other forms of equations relating these parameters to total leaf number have been proposed (Keating and Wafula, 1992; Birch et al., 1998).

Correlation of the k coefficient to the total number of all leaves on the plant was highly positive (r = 0.9229, N = 50, P < 0.05), and the linear regression equation fitted to this relationship was

[8]
where k is the same as in Eq. [5] and n is the total number of leaves.

Equation [8] does not contradict the previous suggestion that the k values may have a biological meaning. As a statistical relationship, Eq. [8] represents only the quantitative relation between k and the total number of leaves, and no more than 85.17% (r2 = 0.8517) of the variation in k values can be accounted for by the total number of leaves. In fact, k values varied considerably within the same total number of leaves (Table 2). The suggestion that k depends on the size of the generative sink is subjective speculation about the physiological relationship. However, this is reasonable because Eq. [8] indicates that the number of leaves should be included in future research efforts to verify the relationships between corn vegetative and reproductive growth.

Although the corn genotypes and growth conditions varied greatly in this data set as shown by the diverse values for the model parameters F0 and k, the product of these two model parameters equaled a constant value of 0.4208 (Fig. 3). The biological meaning of this observation is not clear, but this value may be associated with the partitioning of photosynthate and warrants further physiological research. The maximum leaf area fraction of the corn plants measured in these diverse experimental trials was inversely proportional to the coefficient k values. However, the direct implication in modeling corn leaf area development is that Eq. [5] may be reduced to a two-parameter equation by substitution of k coefficients. This substitution reduces Eq. [5] to the following:

[9]



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  		Fig. 3. The relationship of reciprocal ratio of maximum leaf area fractions (F0) to k in the model of F = F0exp[–0.5(L – L0)2/k2] (Eq. [5]), the relative leaf area distribution model for all mature leaves on corn plants, where F = individual leaf area/plant leaf area, L0 is the downward leaf rank with the maximum area fraction of F0, k is a constant, and r is the correlation coefficient from model fitting.

 
Model Validation
Equations [6] and [7] were used to compute F0 and L0 using the diverse data set from 30 cultivars not utilized in the parameterization of the model. Estimates for plant leaf area were then obtained by dividing observed area of leaf L0 by F0. Comparisons of the model-estimated and measured leaf area values were within 10% for 27 of 30 cultivars (Table 4 and Fig. 4). The correlation coefficient between the estimated and the observed values was 0.9428 (N = 174, P < 0.001). Thus, plant leaf area may be estimated with L0, F0, and leaf area of the largest leaf, and F0 and L0 can be obtained with Eq. [6] and [7] if the number of leaves for a cultivar is known. Since the number of leaves developed on corn plants is nearly constant for specific cultivars grown within a region (Wang, 1999), rapid estimation of corn plant leaf area depends only on the measurement of leaf area of the largest leaf.


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  		Table 4. Mean absolute difference and percentage difference (error) between the predicted and the observed plant leaf area for 30 corn cultivars not used in the model fitting, and the number of sampled plants (PN).

 


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  		Fig. 4. Comparison of observed and estimated total plant leaf area for 30 corn genotypes.

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
The model proposed in this paper can be utilized with relatively high precision (Tables 2 and 3) to describe the individual leaf area distribution on whole corn plants and the green leaf area distribution on corn plants at the silking stage. If dead leaves are counted after the silking stage, the proposed model can be used to calculate photosynthetic leaf area of corn. In addition, area measurements of leaves on a single plant can be fitted to the model (No. 35–53, Table 1; Tables 2 and 3; Fig. 2). Compared with other currently available models (Yang et al., 1998; Dwyer and Stewart, 1986), the proposed model includes fewer parameters with higher stability estimates for the parameters. The representation of individual leaf area distribution on a corn plant is promoted from an empirical level in models to a mechanistic level. The proposed model has the potential to improve rapid leaf area measurement and contribute to computer simulation of canopy leaf area distributions in corn.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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