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MODELING

A Model to Calculate the Vertical Distribution of Grain Number in Pea

^a Unité d'Agronomie, INRA-INAPG, 78850 Thiverval-Grignon, France
^b Unité de Bioclimatologie, INRA, 78850 Thiverval-Grignon, France

jeuffroy{at}bcgn.grignon.inra.fr

	ABSTRACT

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In the pea (Pisum sativum L.) as in most grain legume crops, the seeds are located on the reproductive nodes along the stems. The number of nodes varies widely, and this, combined with variations in environmental conditions during the seed set period, creates a high degree of field-to-field variability in the distribution of seeds along the stems. To model seed number profiles in pea, we adapted a method initially proposed by Dwyer and Stewart for calculating the vertical distribution of plant leaf area in maize (Zea mays L.). The entire profile can be described by two empirical constants, by the number of the individual node bearing the most seeds, and by the maximum number of seeds on one node. These four inputs vary from one location-year to another and are calculated from empirical relationships, taking as explanatory variables the main characteristics of the pea stand. The proposed model simulating seed number per node in pea was evaluated on two samples: one with data from the cultivar used to estimate model parameters (18 points with six different locations and six different years), and another using data from nine other cultivars (27 points). The model gives a reasonable account (r² > 0.80) of the variability in seed number profiles measured in the field. The model uses only one cultivar-dependent parameter (mean weight per seed), and thus it can be easily used by farmers or advisers for practical purposes such as agronomic diagnosis to explain the lack of seeds on some nodes.

Abbreviations: CDD, cumulative degree-days • CGR, crop growth rate • MEP, mean error of prediction • MSEP, mean square error of prediction • NNm, node number with the most seeds in the profile • SNi, seed number on reproductive node i per stem • SNm, maximum number of seeds on one node • SNM2, mean number of stems per m² (branches included) • TRN, total number of reproductive nodes per stem • TSN, total seed number per stem

	INTRODUCTION

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IN PEA, yield varies very widely and is generally closely correlated to seed number per square meter of crop (Hardwick, 1988; Doré et al., 1998). In soybean [Glycine max (L.) Merr.] (Jiang and Egli, 1995), seed number is largely dependent on the crop growth rate during the seed set period. In pea, a close relationship between these two variables has been established for the cultivar Solara and subsequently applied to many other cultivars (Ney, 1994). The seed set period begins at anthesis of the first flower and ends when the pod of the last node has reached the final stage in seed abortion; i.e., the start of seed filling (Ney and Turc, 1993). In pea, seeds are set on several reproductive nodes per plant because of its indeterminate growth habit. These nodes appear successively on the stem. Thus, the youngest pods appear highest on the stem. The total number of reproductive nodes varies widely, both among cultivars and among crop stands sown to one cultivar (Heath and Hebblethwaite, 1987; Turc, 1988; Roche et al., 1998). If too high or too low, the total number of reproductive nodes can be a limiting factor for final seed number and yield (Pate, 1975; Murfet, 1982; Jeuffroy, 1991). A model calculating the mean final number of reproductive nodes per stem in a pea stand, according to the developmental characteristics of the stand, has been proposed and successfully evaluated for numerous environmental conditions and various cultivars (Roche et al., 1998).

Seed distribution among the reproductive nodes of a stem can vary not only according to the number of reproductive nodes, but also according to cultivar, sowing date, and sowing density (Turc, 1988; Jeuffroy and Devienne, 1995). Seed formation does not occur simultaneously on the different plant nodes, because the nodes develop one by one along the stem (for Solara, roughly one every 50 degree-days). Therefore, the number of seeds on each node depends both on the environmental conditions during the seed set period of the node in question (Turc, 1988; Jeuffroy and Chabanet, 1994) and on competition for assimilates among the different sinks present on the plant at this time (Jeuffroy and Devienne, 1995; Jeuffroy and Ney, 1997). This makes it difficult to calculate vertical seed number distribution. However, such estimation of the node-by-node seed number profile is useful from several standpoints; i.e., the profiles can be used as diagnostic tools and for forecasting the effects of limiting factors on final yield.

Few studies have explicitly aimed at analyzing and simulating seed number profiles in pea. The CROP GRO models (Hoogenboom et al., 1992), which were developed specifically for leguminous crops, are based on a population of mean reproductive organs. Though they keep track of individual cohorts over time, they are unable to simulate where on the plant the individual pods and seeds are located. Only one published model (Jeuffroy, 1991; Jeuffroy, 1994a; Jeuffroy and Ney, 1997) allows seed number distribution among the reproductive nodes of a pea plant to be calculated, and it is based on assimilate distribution according to the ratio of supply to requirements from each sink and has been proposed only for the cultivar Solara. However, the parameters of this type of model are cultivar-dependent. Furthermore, it is not easy to adapt these parameters to make the same estimation for other cultivars. The complexity of this model considerably restricts its usefulness under field conditions.

Many studies have been conducted to determine the vertical distribution of plant leaf area, which can be compared to the vertical distribution of seed number per node. Some studies focused on the mean leaf area index of a crop, while others analyzed the vertical distribution of plant leaf area in particular species and provided a basis for simulation (Dwyer and Stewart, 1986, Muchow et al., 1990, and Dwyer et al., 1992, for maize; Muchow and Carberry, 1990, and Carberry et al., 1993, for sorghum; Bouchard, 1997, for wheat). These studies used the same model to calculate the area of individual leaves according to the total number of leaves on the stem and the area of the largest leaf. These two input variables can easily be measured and estimated from empirical relationships (Dwyer and Stewart, 1986; Muchow and Carberry, 1989; Muchow and Sinclair, 1991; Keating and Wafula, 1992; Bouchard, 1997; Fournier and Andrieu, 1998). This model has since been successfully used to compare different cultivars of maize (Dwyer et al., 1992) and of sorghum (Carberry et al., 1993) in which the total number of leaves on the stem and the area of the largest leaf differ widely.

Our objective was to develop a simple model of the vertical distribution of seed numbers on the different nodes of a pea stem, under various environmental conditions and for a range of cultivars. For this purpose, we adapted and evaluated the equations developed for vertical distribution of leaf area to simulate the node-by-node seed number profile on pea plants.

	Materials and methods

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Plant Material and Growing Conditions
Ten field trials were conducted in France between 1989 and 1996 at Chartres, Dijon, Le Rheu, Bignan, Grignon, and Estrées-Mons. Various sowing dates, sowing densities, and varieties were used. The cultivars Finale and Solara, which are known to produce few reproductive nodes with many seeds on the first ones, and the cultivars Alex and Frisson, which generally produce many reproductive nodes with few seeds on each node and the node with the most seeds in the middle of the profile, were used. Basic details of these trials are summarized in Table 1 . Seeds were planted in 30-m rows, spaced 0.175 m apart, in small plots (about 150 m²) randomly distributed within a block and replicated three times.

Sampling and Plant Measurements
Plants were sampled (one sample of 1 m² per plot) at the start of flowering and twice a week from this stage until the final stage in seed abortion of the last pod. At each date, the aerial biomass of each sample was measured after oven-drying at 80°C for 48 h.

At about physiological maturity, one sample (2 m²) per plot was taken at random. The total number of reproductive nodes (TRN) was counted on each stem (branches included) of the sample, and the mean TRN for each treatment was calculated. On each stem, the seeds were counted on each reproductive node, starting with the first. Mean seed number per reproductive node i and per stem (SNi) were then calculated, taking all the stems into account. The total number of stems (branches included) was counted on each sample, and the mean number of stems per m² (SNM2) was calculated for each treatment, taking all the samples into account.

Computational Methods
Time was expressed in cumulative degree-days (CDD) since sowing, using 0°C as the base temperature (Etévé and Derieux, 1982). The number of CDD was calculated as the sum of the mean daily temperatures (average of the minimum and maximum temperatures for each day). Crop growth rate (CGR) during the seed set period was then calculated as the slope of the linear regression of aerial biomass vs. CDD, using the samples taken during the seed set period (duration of 2–3 wk); i.e., about 4 to 6 sampling dates for each treatment.

Only Solara, the cultivar with the most data, was used for model development, and as a consequence, the model was evaluated separately for Solara and the other cultivars. Therefore, the data were divided into three groups. The first set (calibration data) was used for model assessment and parameter estimation with Solara; the second group (Solara validation data) was used to evaluate the models on the same cultivar (Solara) with independent data; and the third group (genotypic extrapolation data) was used to evaluate the model on nine other cultivars. The calibration data set (Table 1) incorporated 25 treatments from the trials at Chartres in 1989 and Grignon in 1994 and 1995. The validation data set included the 18 other treatments with cv. Solara; i.e., Grignon, 1991, 1992, and 1996; Bignan, 1994; Le Rheu, 1994; Dijon, 1994; and Estrées-Mons, 1995. The extrapolation data set consisted of all 27 treatments with the other cultivars; i.e., from the trials at Grignon in 1992, 1995, and 1996 and Estrées-Mons in 1995 (Table 1).

When the model was applied to the other cultivars, the parameters estimated for Solara were not changed. The seed numbers resulting from the simulation were simply adapted to each cultivar by multiplying them by the following ratio: mean weight per seed for cv. Solara/mean weight per seed for the cultivar studied. The mean weight per seed values used for each cultivar were taken from Duchene (1994)(Table 2) .

(1)

where SNi is the number of seeds on the reproductive node i, NNi is the node number, NNm is the node number with the most seeds in the profile, SNm is the maximum number of seeds on this node, and a and b are empirical constants.

As a first step, we fitted Eq. [1] to the values of the profiles measured in the calibration sample, using polynomial transformations in the exponential term, to determine whether the model was well adapted to the shape of the seed number profiles and to estimate the values for variables SNm and NNm and parameters a and b.

Second, by way of multiple linear regressions, we explained the variability of SNm and NNm by reference to the main characteristics of each stand; i.e., SNM2, CGR, TRN, and total seed number per stem (TSN). The data were fitted to several multiple linear regressions, taking into account at the beginning all the first-order interactions. Then nonsignificant terms were withdrawn one by one each time, the least significant one first.

Our last step was to estimate the empirical constants a and b for each location–year–cultivar–density according to the input data (SNM2, CGR, TRN, and TSN) and the variables estimated in Step 2 (NNm and SNm). To do this, we conducted a weighted multiple linear regression of the variable Ln (SNi/SNm) on all individual node data in the calibration data set . We used the square of SNi as a weighting variable (i.e., the inverse of the square of the derivative of the transformation function) as recommended by Aïvazian (1970). Parameters a and b were estimated simultaneously to avoid cumulative errors on each variable. The basic assumptions of the linear models were assessed (normality and independence of the residuals and equality of the variances).

Finally, for evaluation, we determined the node-by-node seed number profile for each location–year–cultivar–density from the validation and the extrapolation data sets (independent data), using each of the proposed equations in succession.

For estimation of model parameters, anomalous data were eliminated from the calibration data set. Measurements were considered anomalous when their Studentized residual >2 in absolute value (SAS Inst., 1987). All data in the validation and extrapolation data sets were used. The models were then evaluated using the mean square error of prediction (MSEP) (Wallach and Goffinet, 1989).

	Results

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Seed Number Profiles and Observed Crop Characteristics
A wide range of node-by-node seed number profiles was obtained from all the location-years. Among and within each data set, the differences between profiles were mainly linked to five characteristics (Table 3) : number of stems per m² (ranging from 33.4 to 257); total number of reproductive nodes (ranging from 2.76 to 12.3); total number of seeds per stem (ranging from 9.73 to 39.01); the node number bearing the most seeds (ranging from 1 to 6); and the maximum number of seeds measured on this node (ranging from 2.20 to 7.16). Crop growth rate during the seed set period also varied considerably among treatments (Table 3), from 5.7 to 21.38 mg m^-2 degree-day^-1.

View larger version (23K): [in this window] [in a new window]   Fig. 1 Comparison of measured and calculated (according to Eq. [1]) numbers of seeds per node per stem for cultivar Solara, grown at two locations in six treatments (treatment codes are given in Table 1). These data sets were used for model development and calibration

When Eq. [1] was applied to calculate the number of seeds on each node for each treatment of the validation sample, giving a simulated seed number profile for each treatment, the calculated profiles for Solara were close to the measured ones. Figure 2 illustrates some treatments chosen from among those that differed most widely in the Solara validation data set.

View larger version (27K): [in this window] [in a new window]   Fig. 2 Measured and calculated node-by-node seed number profiles from some treatments in the validation data set (treatment codes are given in Table 1)

View larger version (26K): [in this window] [in a new window]   Fig. 3 Comparison of measured and calculated values of seed numbers per individual node from all the treatments in the Solara validation data set

View larger version (28K): [in this window] [in a new window]   Fig. 4 Measured and calculated node-by-node seed number profiles from some situations in the extrapolation data set of cultivars other than Solara (treatment codes are given in Table 1)

View larger version (29K): [in this window] [in a new window]   Fig. 5 Measured and calculated values of seed numbers per individual node from all the treatments in the extrapolation data set of cultivars other than Solara

	Discussion

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The two parameters and the two input variables needed for the model were estimated, with varying degrees of precision, from measured crop characteristics (SNM2, CGR, TRN, and TSN) that are easy to measure or calculate. For instance, the number of stems per square meter was easy to measure at the start of flowering, and was, in any case, systematically measured in trials conducted for recording references. Crop growth rate during the seed set period could be calculated from a growth model such as the one described and calibrated by Jeuffroy and Ney (1997). This model is based on the formulation by Monteith (1972, 1977), assuming that radiation interception by the crop was at its maximum after flowering began, and using weather data on incident radiation as inputs. The period for estimating CGR (cellular divisions phase) could be determined from the developmental models developed by Ney and Turc (1993) and Roche et al. (1998). The total number of reproductive nodes could be calculated using the model proposed by Roche et al. (1998). Finally, the total number of seeds per stem could be simulated from CGR, using the relationship proposed by Ney (1994).

The physiological basis of the effect of the explanatory variables in the linear models simulating SNm and NNm could also be inferred. In Eq. [2] (Table 5), for a mean value of TRN (on the calibration sample, the mean value was 7.42), the coefficient affecting SNM2 was positive. The result, that NNm was located higher on the stem when SNM2 increased, was consistent with the observations of Turc (1988) that when stem density was higher, incident radiation penetrated less effectively into the crop and seed abortion toward the bottom of the profile was more frequent. The coefficient affecting TRN was positive, which would also be predicted (the higher the TRN, the higher the probability of a high value of NNm). The maximum number of seeds on one node depended on TSN, CGR, and TRN (Eq. [3] in Table 5). The effect of TSN was, of course, positive. The effect of TRN was negative, which meant that the maximum number of seeds on one node was lower on plants bearing more nodes. This result was consistent with the fact that the same total number of seeds were shared among more nodes, so each node was likely to bear fewer seeds. The negative effect of CGR was probably due to interactions with the other two variables. The equations used to simulate the model's inputs and parameters were fairly simple, attesting to the robustness of these effects.

The model calibrated on Solara was extrapolated to other cultivars using only the ratios of the mean weights per seed for each cultivar, an easily measured characteristic of each cultivar (Duchene, 1994). The evaluation of this adapted model on the 10 other cultivars studied did not give any systematic bias linked to this factor. The anomalous data resulting from situation G95-Al1 led us to exclude from the model's range any crop with more than 12 reproductive nodes, an extremely rare occurrence in typical field conditions. Therefore, the proposed models give a reasonable account of the measured variability in seed number profiles within a cultivar and also among cultivars, without having to introduce a large number of cultivar-dependent parameters. This result increases the relevancy of these models under field conditions, and makes them complementary of more mechanistic and complex models.

Some of the trials used in this study were affected by stresses during seed set. For example, we measured water stress in treatment G95-S1NI, as the soil water potential, a factor limiting seed set (Ney et al., 1994), was below -0.06 MPa from the day following the start of flowering. Several days with daily mean air temperature >25°C occurred during the seed set period in treatments G95-S2, a factor known to induce seed abortion in pods (Jeuffroy et al., 1990). Finally, a compact soil structure was measured in treatment G94-S1-90T, a factor which generally affected N fixation by the root nodules (Tricot et al., 1990; Doré, 1992). Even in these environments, the models allowed us to simulate effectively the number of seeds per individual node, attesting to the robustness of the model. However, we did not investigate all possible limiting factors (e.g., diseases), and the number of treatments used for each of the tested factors was generally too low to draw any conclusion as to the validity of the model in these cases. To determine the range of validity of the model more precisely, one would have to conduct a more systematic verification of the model's stability for the main limiting factors on pea crops, as encountered in farmers' fields (Doré et al., 1997; Doré et al., 1998).

It is fairly easy to incorporate this model (i.e., Eq. [1], [2], [3], [4], and [5]) into a more complete crop simulation model, since, as we have shown, most of its input variables can be simulated. At present, only the number of stems per square meter, which varies widely according to plant density and to branching rate (Doré et al., 1998), cannot be simulated.

The model is especially useful because it is easy to apply and easy to extrapolate to other cultivars. For instance, as Pigeaire (1986) has shown with soybean, seed number profiles can be a useful and effective tool for diagnosing the factors that limit the plant's reproductive development, such as high temperatures (Jeuffroy, 1994b) and water stress (Ney et al., 1994). However, given the precision of the model, only the main effects could be differentiated from simulation errors. On the other hand, these profiles can be a tool for forecasting the impact of limiting factors, such as aerial disease or water stress, on final seed yield. Depending on the date of their occurrence relative to the seed set period of each node, seed number could be reduced to a greater or lesser extent on some nodes, and the overall effect on seed yield could be nil (Béasse, 1998) or great (Ney et al., 1994).SAS Institute 1987

	ACKNOWLEDGMENTS

 
We sincerely thank B. Ney (INRA Dijon), B. Tivoli (INRA Le Rheu), and Martine Duparque (GIE pois, Estrées-Mons) for supplying us with their own data. The trials at Grignon were conducted by the Unité Expérimentale INRA, and the trial from Chartres was grown by a farmer—our sincere thanks to all concerned. Financial support came from the Union Nationale Interprofessionnelle des Plantes Riches en Protéines (UNIP). We particularly thank R. Bonhomme for his advice on earlier drafts of this paper, and Harriet Coleman for the English revision.

Received for publication January 4, 1999.

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