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Predicting potential kernel set in maize from simple flowering characteristics. (Crop Physiology & Metabolism).

THE DEVELOPMENTAL PROCESSES that determine kernel number per plant in maize can be divided into three consecutive stages for simulation purposes. In the first stage, the reproductive structures are initiated and differentiated into staminate (male) flowers on the apical inflorescence (tassel) and pistillate (female) flowers on one or more lateral inflorescences (ears). In the second stage, the flowers are functionally mature and engage in pollination; male flowers release pollen grains from the anthers and female flowers exsert receptive stigmas (silks) to intercept the airborne pollen. Separate male and female flowers require that the timing of pollen shed coincide closely with silk exsertion and that tassels produce an overabundance of pollen to ensure pollination of exposed silks. In the third stage, pollination is followed by fertilization and kernel formation (Kiesselbach, 1999). Current models for simulating maize yield do not consider the quantitative and dynamic nature of the first two of these flowering stages.

Most efforts to simulate kernel formation in maize have attempted to associate the final kernel number per plant with the current supply of photosynthate or related characteristics such as light interception or plant growth rate around the time of silking (Edmeades and Daynard, 1979; Tollenaar et al., 1992; Andrade et al., 1993; Kiniry and Knievel, 1995; Otegui, 1997; Andrade et al., 1999; Andrade et al., 2000). Edmeades and Daynard (1979) were the first to relate the number of kernels per ear with a calculated rate of photosynthesis per plant at anthesis. They used a rectangular hyperbola to describe this relationship, which was the basis for predicting kernel number per plant in the original version of CERES-Maize (Jones and Kiniry, 1986).

Tollenaar and colleagues (1992) found that kernel number per plant was related to the rate of dry matter accumulation 1 wk before to 3 wk after silking. They observed also that modern hybrids tended to produce kernels on a second (subapical) ear at low plant population densities. Therefore, they proposed a double-hyperbola function to describe the relationship between kernel number per plant and plant growth rate. Otegui (1995), however, reported that among Argentinean hybrids, only prolific types tended to produce kernels on subapical ears at commercial plant densities.

Because canopy photosynthesis is controlled primarily by solar radiation (Christy et al., 1986), a similar relationship between kernel number and light interception would be expected. Andrade et al. (1993) reported that the number of kernels per plant was curvilinearly related to the IPAR per plant during a 31-d period centered on silking. Kiniry and Knievel (1995) used a linear equation to relate kernels per plant with IPAR during 10-d period following silking, as did Otegui (1997), for IPAR during a 30-d period around silking. Andrade et al. (2000) and Lizaso et al. (2001), however, observed that curvilinear functions predicted kernel numbers from IPAR more accurately than did a simple linear function over a broad range of plant growth rates.

There is evidence that kernel number per ear is not simply a function of assimilate supply. Struik et al. (1986) associated the effects of temperature and photoperiod on kernel number with asynchrony between pollen shed and silk emergence. Because these environmental variables are known to alter floral development and function (Herrero and Johnson, 1981; Hall et al., 1982; Struik et al., 1986; Bassetti and Westgate, 1993c), a mechanistic description of the floral biology of maize may be needed to improve forecasting of kernel number per plant across a wide range of environmental conditions. Few studies provide the quantitative information needed to relate floral development and kernel formation.

Bassetti and Westgate (1994) measured kernel number per ear formed at various intensities of pollen shed.

In their study, seasonal pollen shed rates followed a normal distribution that peaked at [approximately equal to] 500 grains [cm.sup.-2] [d.sup.-1] two days after anthesis (50% of population shedding pollen). They also determined that daily rates of pollen shed densities at 100 grains [cm.sup.-2] [d.sup.-1] or greater reaching the plane of exposed silks were sufficient to achieve maximum kernel number per ear. Struik and Makonnen (1992) reported a similar pattern of pollen shed, but did not relate it to kernel number.

Bassetti and Westgate (1993a,b) described the dynamics of silk emergence and senescence on individual ears of maize as a progressive process that varies by genotype and environmental conditions. This silk exsertion dynamic for individual plants provides the essential link between the population dynamics of silking (percentage of plants with silks exposed) and the number of silks exposed for pollination on an area basis, which ultimately determines the potential for kernel set.

These few reports on pollen shed and silk emergence indicate the processes of male and female flowering in maize progress in a fairly predictable manner. As such, it should be possible to develop a mechanistic description of kernel set on a field scale based on a quantitative evaluation of synchrony between pollen production and silk exsertion. Sadras et al. (1985a,b) recognized this possibility in their work, but predicted kernel set only on a relative, basis. They also lacked sufficient quantitative information on the flowering and pollination processes to predict kernel number when pollen amount was limiting. The objective of this study, therefore, was to describe mathematically the processes of floral anthesis (i.e., pollen shed and silk emergence) and pollination. In this initial effort, treatments were designed to alter kernel set by limiting pollen amount. The potential impact of environmental limiting conditions on floral synchrony and pollen viability were not considered, but procedures were developed to accommodate the effects of such constraints.

MATERIALS AND METHODS

Procedure for Estimating Kernel Number per Plant

The procedure for estimating kernel number per hectare is based on a quantitative description of maize flowering characteristics on a field scale. Mathematical functions were fit to temporal profiles of plant population dynamics for pollen shed and silk exsertion measured in the field. The characteristics described are: (i) amount and temporal distribution of pollen shed, (ii) amount and temporal distribution of exserted silks; (iii) relationship between kernel set and daily pollen shed intensity reported by Bassetti and Westgate (1994). These components were linked mathematically to estimate the total number of receptive silks pollinated each day. This value represents the potential number of kernels that can be formed by the plant population.

Amount and Temporal Distribution of Pollen

We used the field data of Westgate et al. (2003) for pollen shed in different MF treatments to generate a seasonal distribution of pollen production per plant. Daily values for pollen shed (grains [cm.sup.-2]) were normalized to daily pollen shed per fertile plant by correcting for the fraction of MF plants and the population density in each treatment. Gauss curves were fit to these normalized data to generate a genotype-specific pattern of seasonal pollen shed given by:

[1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where PR is the rate of pollen shed on a fertile plant basis (grains per plant per day), p is the total amount of pollen produced per plant (grains per plant), W is the width of the pollen shed curve measured at half the maximum pollen shed rate (days), and t and [t.sub.x] are the current day and the day of maximum pollen shed. Rates of pollen shed per plant predicted from the Gauss curves were scaled to rates of pollen shed on an area basis (grains [cm.sup.2] [d.sup.1]) using the population densities and fraction of fertile plants in each MF treatment.

The seasonal distribution of pollen shed also was predicted from field observations of male flowering characteristics at the population level. The three stages of tassel development identified in Westgate et al. (2003): beginning of pollen shed (start shed--first anthers visible on main tassel branch), maximum intensity of pollen shed (max shed--all branches of tassel are involved in pollen shedding), and ending of pollen shed (end shed--no new anthers visible) were expressed as a percentage of the plant population on a daily basis. The progress of each stage was described using a sigmoid logistic function:

[2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where Pop and [Pop.sub.x] are the percentage of the plant population at each stage of pollen shed and the corresponding maximum percentage; t and [t.sub.m] are the current day and the day when 50% of the plant population reaches each stage of pollen shed; k is a curve parameter governing the slope of the function.

Using the predicted percentage of plant population at each stage of pollen shed (Pop), a [P.sub.ind] was calculated to simulate the daily rate of pollen produced in the field:

[3] [P.sub.ind] = Start shed + Max shed/2 - End shed,

where [P.sub.ind] is the population index (%), and start shed, max shed, and end shed are the predicted percentages of the plant population that have begun to shed pollen, reached maximum shed intensity, or completed pollen shed. Calculated [P.sub.ind] values were scaled to daily rates of pollen shed on an area basis (grains [cm.sup.-2] [d.sup.-1]) using the population density and the total pollen production per plant. Details of these procedures and calculations are provided by Westgate et al. (2003).

Amount and Temporal Distribution of Receptive Silks

The distribution of silks exposed for pollination in the field results from developmental processes occurring at the plant level and at the population level. At the plant level, silks emerge from the surrounding ear leaf sheaths (husks) during a period of days following a curvilinear pattern (Bassetti and Westgate, 1993a, 1994). We used a monomolecular model to describe the pattern of silk exsertion on each ear:

[4] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where SN is the cumulative number of exposed silks on one ear, S[N.sub.x] is the potential number of silks per ear equal to the potential number of florets, [t.sub.0] is the time when the first silk is exserted (day of year), and b is a shape parameter controlling the slope of the curve.

At the population level, the percentage of plants with silks exposed progresses in a sigmoid fashion. We used a sigmoid logistic function, similar to Eq. [2], to characterize this developmental pattern. Daily percentages for each treatment were converted to number of plants silking by multiplying by the plant population density (plants [ha.sup.-1]).

The number of newly exserted silks per unit land area was calculated each day by multiplying the pattern of silk exsertion by individual plants times the pattern of silk emergence for the population. The daily cohort of plants that began to exsert silks was calculated from the population dynamics curves for each treatment. Measurements of silk emergence patterns for the hybrids used in this study indicated silks continued to emerge for up to 9 d on unpollinated ears. Therefore, it was assumed each plant assigned to a daily cohort would exsert silks for the next 9 d (Bassetti and Westgate, 1993a) according to Eq. [4]. The same calculation was made for each subsequent day until 100% of the plants in each treatment were silking. The number of new silks exserted on an area basis was calculated each day as the sum of silks exserted by each of these nine cohorts.

An alternative mathematical approach used to describe the daily emergence of silks was a double Gauss function fit to the seasonal pattern of newly exposed silks in the field:

[5] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [S.sub.tot] is the daily number of newly exposed silks per land area, [P.sub.1] and [P.sub.2] are the area of the Gauss Curves 1 and 2 whose addition represents the total number of exposed silks (or viable flowers) during the season per unit area, [W.sub.1] and [W.sub.2] are shape parameters controlling the width of the curves (days), and [t.sub.x1] and [t.sub.x2] are the times when the peaks of Curves 1 and 2 are reached (days).

Relationship between Kernel Set and Daily Pollen Shed Intensity

We applied two linear functions in series to the published data of Bassetti and Westgate (1994) to calculate a percentage of kernel formed by flowers with exposed silks from the daily rate of pollen shed. The limits of pollen shed rate for each equation were:
[6] ks = 0.96 x pr   0 < = pr [less than or equal to] 100, and
    ks = 96          pr >  100.


Where ks is the percentage of florets with exposed silks that formed kernels (%), and pr is the daily rate of pollen shed (grains [cm.sup.-2] [d.sup.-1]). In this formulation, 100 grains [cm.sup.-2] [d.sup.-1] is the critical rate of pollen shed below which ks is less than the maximum 96%. Because Bassetti and Westgate (1994) used final kernel set in their analysis, their equations relating pollen shed density to percentage of kernel formed incorporate the natural level of pollen viability inherent in their experiment. Nonetheless, our approach provides the mathematical basis for testing the impact of pollen viability on kernel number per hectare by adjusting the critical rate of pollen shed required for maximum ks.

Daily kernel formation as measured by Bassetti and Westgate (1994) never reached 100% of exposed silks even at the highest rates of pollen shed (Eq. [6]). Kernel abortion was nil in their study since they measured kernel set only at highly receptive floral positions in the middle of the ear. Therefore, we assumed the remaining florets with exposed silks remained receptive. These silks were added to the next day's pool available for pollination. These unpollinated silks remained receptive to pollen for five additional days. Silks that were not pollinated by the sixth day were assumed to senesce and lose receptivity (Bassetti and Westgate, 1993b) and no longer contributed to kernel set.

Evaluation of the Procedure for Estimating Kernel Set

We compared the number of florets per hectare and kernels per hectare calculated by the model with measured floret and kernel numbers in eight isolated field plots that varied widely in pollen shed density. In 1986, MF levels of 100, 50, 20, and 0% were obtained by detasselling a 100% MF hybrid, Pioneer 3978 (hereafter P3978). In 1987, MF levels of 75, 50, 20, and 0% were obtained by mixing MS and MF isolines of Pioneer 3925 (hereafter P3925). P3978 was planted at 7.2 plants [m.sup.-2], and produced an average of 658 silks per ear. P3925 and the pollen sterile isoline P3925S were planted at 5.7 plants [m.sup.-2], and produced an average of 621 silks per apical ear. Details of the experimental design and plant culture conditions are provided in the companion paper by Westgate et al. (2003).

In the model, each MF treatment was corrected for plant population density and average number of ears per plant at harvest. Percentage of plants at each stage of pollen shed or reaching silking was recorded every other day by evaluating at least 80 plants in the four center rows of each plot.

Procedures used to document the daily rate of pollen shed and daily progress of silk exsertion are detailed in Bassetti and Westgate (1994) and Westgate et al. (2003). Briefly, passive pollen traps were placed within the canopy at ear level and collected every 24 h at [approximately equal to] 1700 h for counting. Pollen density on the trap surface was evaluated in the laboratory by image analysis according to Bassetti and Westgate (1994). Every 2 d beginning at first silk appearance, 1 to 2 cm of unpollinated silk tissue was collected from ears of 10 plants. Care was taken to collect all exposed silks and to prevent pollination at all other times by covering the ears with a glassine bag. Silk pieces were fixed in ethyl alcohol-glacial acetic acid-formalin-water (50:5:10:35 by volume) containing 10 g [kg.sup.-1] analine blue, and subsequently counted by hand.

Effect of Asynchronous Pollination Within-Ear Synchrony

Several studies indicate that kernel set at flower positions with late emerging silks increases dramatically if all exposed silks on the ear are pollinated synchronously (Freier et al., 1984; Carcova et al., 2000). Evidently, both the timing of pollination and the number of prior fertilization events affects kernel set by later pollinated flowers. Therefore, the effect of asynchronous pollination of florets on the apical ear was included following the hypothesis that earlier-formed kernels could have a detrimental effect on the success of kernel formation by later-fertilized flowers. The daily calculation of kernel set (Eq. [6]) was modified to account for the cumulative number of kernels set per ear (KS) and asynchronous pollination efficiency ([E.sub.AP]):

[7] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [E.sub.AP] is a unitless function (0 to 1) that varies with the cumulative number of kernels per ear formed the previous day (KS). Parameter c controls the slope, and K[S.sub.0] is the intercept for kernel set. If [E.sub.AP] = 1.0, all pollinated flowers will set kernels. If [E.sub.AP] = 0.0, no pollinated flowers will set kernels, regardless of the pollen shed density. Parameters in Eq. [7] (c = 0.013; K[S.sub.0] = 1.2 x S[N.sub.x], for apical ear; K[S.sub.0] = 1.1 x S[N.sub.x], for subapical ear) were calibrated to data in Carcova et al. (2000). Asynchrony-adjusted kernels set was calculated daily using Eq. [8]:

[8] KS = SN x ks x [E.sub.AP],

where KS is the adjusted kernel set (kernels [ear.sup.-1]), SN is the cumulative number of silks exposed, ks is the fraction of kernel set estimated with Eq. [6] (%), and [E.sub.AP] is the efficiency factor for kernel set calculated inn Eq. [7].

Between-Ear Synchrony

We included the effects of asynchrony between apical and subapical ears of P3925 in terms of delayed silk exsertion and a decreased number of flowers per ear on subapical ears. P3978 was not included in this analysis because it did not produce second ears. Silk exsertion on subapical ears of P3925 was estimated from the pattern of silk exsertion measured on apical ears and the measured relationships between apical and subapical ear development in a similar prolific hybrid (Table 1). the experiment providing data on flowering of subapical ears was planted at the Bruner Research Farm near Ames, IA, in 2000. Silk exsertion was evaluated on the prolific hybrid Asgrow 740 grown at three population densities (1, 8, and 16 plants [m.sup.-2) supplied with 168 kg N [ha.sup.-1] and replicated three times in a split-plot design. When 30% of the plants in each plot reached silking, five plants about to exsert silks were selected for sampling. Ears were covered with glassine bags to prevent pollination, and silks were sampled 2, 4, and 8 d after first silks appeared. Two-centimeter segments of silk tissue were cut at husk level, transferred to plastic bags containing 500 g [kg.sup.-1] ethanol, and stored at 4[degrees]C until counted manually.

Monomolecular functions (Eq. [4]) were fit to measured silk exsertion data for the apical ear of P3925 and for apical and subapical ears of Asgrow 740 (Fig. 1). Parameters SN (maximum silk number), b (shape factor), [t.sub.o] (initial day of silk emergence), and duration of silk emergence were generated from each curve. Parameters for subapical ears of P3925 were calculated relative to the apical ear using the ratio of apical/subapical values for the corresponding Asgrow 740 parameters. Subapical ear parameters for P3925 were adjusted to 5.7 plants [m.sup.-2] by linear interpolation. The delay in initial silk emergence for the subapical ear was calculated as [t.sub.o2] - [t.sub.o1].

[FIGURE 1 OMITTED]

Prediction Evaluation

Deviation of predicted and observed values were compared using the root mean square of error (RMSE):

[9] RMSE = [[1/n [[summation of].sup.n.sub.i=1] [([P.sub.i] - [O.sub.i]).sup.2]].sup.0.5],

where [P.sub.i] and [O.sub.i], are predicted and observed values, and n is the number of observations.

RESULTS

Dynamics of Pollen Shed

The first step in the procedure to calculate the number of kernels formed on an area basis was to characterize the dynamics of pollen production in the field. The evaluation of daily pollen production showed greater variability of daily rates for P3978 than for P3925 (Fig. 2a and 2b). Also, there was a greater amount of pollen drift into the 0% MF treatment for P3978. These results likely reflected differences in methodology used to reduce the levels of pollen production (detasselling in P3978 vs. MS isoline in P3925). The difference in plant population density (7.2 plants [m.sup.-2] for P3978 vs. 5.7 plants [m.sup.-2] for P3925) contributed to the lesser maximum rates of pollen shed for P3925 as well. To compare the response of two hybrids to pollen shed density, pollen shed rates were normalized using plant population density and the fraction of MF plants. Normalized in this way, it was evident that the MF treatments did not affect the pattern or the timing of the pollen shed process (Fig. 2c,d). Pollen production per plant based on the seasonal pollen shed curves fit to the field data were very similar for the two hybrids: 4.2 million grains per plant for P3978 and 4.5 million grains per plant for P3925. The seasonal pattern of pollen shed, however, was 4 to 5 d longer for P3978. The normalized curves for pollen production per plant were scaled to pollen shed rates on an area basis (grains [cm.sup.-2] [d.sup.-1]) using the population densities and relative seasonal pollen production for each hybrid-treatment combination (Fig. 2a,b). These scaled curves of pollen production accurately described the seasonal pattern of pollen shed for each treatment and were used to predict kernel number per hectare.

[FIGURE 2 OMITTED]

The pollen trap method accurately characterizes the dynamics of pollen shed in the field (Bassetti and Westgate, 1994; Fonseca et al., 2002). But, it is a labor and time-intensive process. Therefore, an alternative procedure to describe the seasonal pattern of pollen shed in the field was needed. Westgate et al. (2003) showed that the daily rate of pollen shed could be derived from simple observations of the male flowering process. Using their procedure, we fit sigmoid curves (Eq. [2]) to field observations of percentage of plants beginning pollen shed (anthesis), reaching maximum shed, and ending shed (Fig. 3a,b). Simulated percentages of population at beginning, maximum, and ending pollen shed were used to calculate a [P.sub.ind] of plants shedding pollen (Eq. [3]). To simulate the intensity of pollen shed, the [P.sub.ind] was scaled using the total pollen production per plant (parameter p in Eq. [1]) and the plant population density (Westgate et al., 2003). Figure 3c and 3d show that the rates of pollen shed computed from the [P.sub.ind] were in close agreement with rates measured in the field. Values of RMSE were similar to corresponding values obtained when rates of pollen shed were predicted with the scaled Gauss curves (Fig. 2a,b). These results indicate that the [P.sub.ind] procedure provides a promising alternative to the pollen trap method to characterize the dynamics of pollen shed in the field. If the average pollen production per plant is known, pollen production on an area basis can be distributed realistically in time using simple observations of male flower development.

[FIGURE 3 OMITTED]

Dynamics of Silk Exsertion

The second component required to calculate kernel number per hectare is a measure of the dynamics of silk numbers available and receptive to pollination. This dynamic requires a simultaneous calculation of silks exserted per plant and an estimate of the percentage of plants beginning to exsert silks. We calculated silk numbers per hectare first on a cumulative basis as depicted in Fig. 4a and 4b. The process is described by Eq. [4], which accounts for variations in the maximum number of florets per ear (S[N.sub.x]) and the duration of silk exsertion on an ear (days). The hybrids examined in this study required 8 and 9 d to complete silk exsertion, which is in agreement with Bassetti and Westgate (1993a), who reported differences up to 4 d between hybrids to complete the silking process.

[FIGURE 4 OMITTED]

To calculate the number of newly emerged silks per hectare, we computed the daily components for each plant and for the population (Fig. 4c and 4d). Each day, a new group of plants begins to exsert silk simultaneously with a second group in its second day of silk exsertion, along with those in their third day, and so on. After calculating the number of silks exserted by each group based on their respective day of silking, the values are added to obtain a daily total of newly exposed silks (Fig. 5). A double Gauss model also described the dynamics of silk exsertion for each hybrid. The model explained >99% ([r.sup.2] = 0.999) of the seasonal variation in silk exsertion per hectare. The maximum rate of silk appearance occurred 2 to 3 d after silking for the population (50% of population with silks exposed), which corresponds closely with the maximum rate of pollen shed for both hybrids. On a population basis, the two hybrids exposed silks during a period of 31 to 34 d (Fig. 5). By the day of silking, the populations had exserted 20 to 25% of their seasonal totals.

[FIGURE 5 OMITTED]

Pollen and Silks Synchrony: Potential Kernel Set

Each exposed silk translated mathematically into a potential kernel, depending on its timely pollination. So the third component in our predictive procedure was to link the dynamics of silk exsertion with the dynamics and intensity of pollen production to generate a cumulative curve of kernel set for each treatment. We relied on two facts about the flowering biology of maize to estimate potential kernel set: (i) kernel set varies with pollen shed density in a manner defined by Bassetti and Westgate (1994); and (ii) each exposed silk remains receptive to pollen for at least 6 d (Bassetti and Westgate, 1993a).

Each day, a new group of silks was exposed for pollination, a certain rate of pollen shed was available to pollinate them, and the daily pollen rate was associated with an expected percentage of fertilized florets (Bassetti and Westgate, 1994). To calculate the number of kernels set each day, we multiplied the number of silks exposed that day times the percentage of florets that should be fertilized given the daily rate of pollen shed (Fig. 6). Figure 7 shows the daily progress of kernel addition for two levels of pollen availability provided to P3978 and P3925. The model predicted that hybrid P3978 would pollinate 96% of the exposed silks when 100% of the plants contributed to pollen shed (Fig. 7a). All exposed silks were successfully pollinated until the daily rate of pollen shed decreased to [approximately equal to] 100 grains [cm.sup.-2]. Reducing the number of plants shedding pollen to 20% MF decreased the seasonal production of pollen to 30% of the 100% MF rate. Yet, sufficient pollen was available to fertilize [approximately equal to] 90% of the flowers with exposed silks (Fig. 7c).

[FIGURES 6-7 OMITTED]

Hybrid P3925 apparently was not as efficient as P3978 in pollinating receptive silks. With 75% of the plants shedding pollen, only 86% of the florets with exposed silks set a kernel (Fig. 7b). When the fraction of MF plants decreased to 20% MF, seasonal pollen production was reduced to 24% of the maximum, and 71% of flowers with exposed silks produced kernels (Fig. 7d). This supports the notion that pollen production is generally well in excess of pollination requirements for >90% of exposed silks. Maize canopies might fail to pollinate late emerging silks, however, even when all plants are MF.

Calculated values for silking florets per hectare and kernels per hectare were compared with field measured floret and kernel numbers. Our procedure predicted floret numbers with >99.5% accuracy in all treatments, when the specific plant population and average number of ears per plant were considered (Table 2). Predicted kernel numbers were overestimated, however, by up to 15% for P3978 and up to 20% for P3925 even in the 20% MF treatment, which was shown to limit kernel set (Westgate et al., 2003). This consistent overestimation of kernel number suggested that a systematic factor affecting kernel set was not considered in the model.

Pollen Viability

The relationship of Bassetti and Westgate (1994) adopted to predict kernel set (Eq. [6] and Fig. 6) included an unknown level of pollen viability. By using 100 grains [cm.sup.-2] [d.sup.-1] as the critical pollen density to assure 96% kernel set in Eq. [6], we tacitly assumed that the level of pollen viability in Bassetti and Westgate (1994) applied to P3798 and P3925. A lower level of pollen viability in our experiment might explain the overprediction of kernel numbers in Table 2. To test this possibility, pollen viability was decreased mathematically by increasing the critical pollen rate in Eq. [6] to values >100 grains [cm.sup.-2] [d.sup.-1]. For this evaluation, we selected MF treatments in which pollen production was similar for both hybrids and was low enough to limit kernel set: the 20% MF treatment for P3978 and the 50% MF treatment for P3925 (Westgate et al., 2003). Table 3 shows that decreasing pollen viability lowered kernel set in these treatments, but the impact was too small to account for the overprediction observed, even when pollen viability was decreased by 37%, which is typical of high temperature stress (Herrero and Johnson, 1980) or field aging (Luna et al., 2001).

Pollination Synchrony within Ear and between Ears

The initial calculation of potential kernel set assumed that the opportunity of a receptive silk to be pollinated and develop into a kernel depended only on the rate of pollen shed. It also assumed that silk exsertion on apical and subapical ears were identical in terms of number and timing. Yet Carcova et al. (2000) showed that naturally pollinated plants often set fewer kernels than did plants whose silks were pollinated synchronous by hand, particularly at low population densities. Also, silk emergence on subapical ears often is delayed relative to apical ears (Jacobs and Pearson, 1991; Otegui and Melon, 1997), with greater delays evident at higher plant population densities (Jacobs and Pearson, 1991; [Carcova] et al., 2000).

To take these developmental effects into account, we modified the procedure for calculating kernel number to include the delay in silk exsertion by subapical ears and the effects of asynchronous pollination within and between ears on the same plant (Fig. 1, Table 1). Although these refinements are based on existing evidence in the literature, we did not measure these developmental effects directly in our data set. Therefore, they are considered as hypothetical effects in our analysis that will require experimental confirmation.

Figure 8a shows that our procedure correctly forecast the lower potential of P3925 florets to form kernels (83% maximum set) compared with those of P3978 (65 % maximum set) at high pollen shed density. Including the effects of asynchronous pollination and the delayed development of subapical ears decreased kernel set (relative to the number of exposed silks) substantially in both hybrids (Fig. 8b). Predicted kernel set closely matched measured kernel set at all pollen shed densities for P3978, but the procedure only accounted for about half of the overestimate in P3925.

[FIGURE 8 OMITTED]

DISCUSSION

The purpose of this work was to develop a model to simulate kernel set from the mathematical description of male and female flowering dynamics in a maize field. We attempted to generate general algorithms useful to improve existing models of kernel number (Lizaso et al., 2001). The functions were developed for nonstress conditions, but can be adapted readily to accommodate a range of conditions known to alter flower development and function. Rates of pollen shed density measured at the level of exposed silks were accurately described with Eq. [1] (Fig. 2a,b). Maximum deviation between predicted and measured pollen rates (as expressed by the RMSE) was 92 grains [cm.sup.-2] [d.sup.-1] for P3978 at 100% MF. Similar rates of pollen shed were reported earlier (Hall et al., 1982; Struik and Makonnen, 1992; Bassetti and Westgate, 1994). We also calculated daily appearance of silks using Eq. [4] (Fig. 4) following patterns in previous reports (Bassetti and Westgate, 1994; Carcova et al., 2000). These patterns and timing of pollen shed and silk appearance also can be modified readily to predict kernel set under stress conditions. High temperature from tassel initiation to kernel set, for example, shortens the duration of pollen shedding (Struik et al., 1986). Drought at or before tasseling delays silk emergence (Herrero and Johnson, 1981; NeSmith and Ritchie, 1992). Flooding during the early vegetative growth delays silking more than tasseling (Lizaso and Ritchie, 1997). These environmental responses can be simulated by altering parameters in Eq. [1] and [4].

Our calculations show that the seasonal duration of silk exsertion for the eight populations examined persisted for a period of 31 to 34 d (Fig. 5). The maximum duration of pollen shed was 24 d for P3978 and 17 d for P3925. Rates of pollen shed at or above that required for maximum kernel set (Fig. 6), however, occurred only on eight of these days for P3978 and on five days for P3925. Because silks remain receptive to pollen for 6 d after they first emerge (Bassetti and Westgate, 1994), most of the unpollinated silks exserted early in pollen shedding were pollinated eventually (Fig. 7). The beneficial impact of this prolonged silk receptivity was most evident in the 20% MF treatments in which a large reduction in pollen deposition early in silking resulted in only a 6% (P3978) and 15% (P3925) reduction in kernel set. Silks exserted after the rate of pollen shed was less than required for maximum kernel set constituted the vast majority of unpollinated silks. In this regard, the longer duration of pollen shed for P3978 (Fig. 2c) contributed to its advantage over P3925 in terms of setting a larger proportion of kernels per exposed silk (Fig. 8).

In the treatments that were saturated for seasonal pollen deposition ([greater than or equal to] 50% MF), 83% and 65% of the pollinated florets developed into kernels for P3978 and P3925, respectively (Table 2). These percentages are in general agreement with reported values (Otegui and Melon, 1997). Yet, our calculations overpredicted the percentage of florets that set kernels both under pollen-abundant and pollen-limited conditions. When pollen is abundant, some degree of overprediction is expected since kernel set is limited by other factors, such as available assimilate supply (Andrade et al., 1999; Andrade et al., 2000; Lizaso et al., 2001). Under pollen-limited conditions ([approximately equal to] 30% of maximum pollen shed for each hybrid), however, overprediction of kernels per hectare indicates that the model did not consider one or more important factors constraining kernel formation.

We tested the possibility that the level of pollen viability was lower than originally assumed in the Bassetti and Westgate (1994) relationship between pollen shed density and percentage kernel set. Decreasing the efficiency of kernel set at low pollen shed density by 37% (effectively decreasing pollen viability) decreased predicted kernels per hectare by <5% in MF treatments that were already marginal for pollen amount (Table 3). Therefore, it seems unlikely that this factor alone could account for the overestimation of kernel set. This exercise, however, demonstrates the utility of our modeling approach to quantify developmental and environmental effects on kernel set.

A second possibility was that the original calculation of kernel number did not consider the potential for kernel loss due to asynchronous pollination within and between ears. We tested two related components of asynchronous pollination for their impact on kernel set. First was the developmental dominance of basal kernels over apical kernels on the same ear (i.e., within-ear asynchrony). The second was the delayed development of subapical ears relative to apical ears (i.e., between-ear asynchrony).

Carcova et al. (2000) showed that synchronously hand-pollinated ears set more kernels than did naturally pollinated ears at the same level of assimilate supply. On the basis of their results, we developed an asynchronous pollination efficiency factor ([E.sub.AP]) to account for the dominance of early-formed kernels on predicted kernel number. Originally, the opportunity for a silk to be pollinated and for its floret to develop into a kernel depended only on the daily rate of pollen shed as dictated by Eq. [6]. In the modified procedure, the fraction of kernels set each day decreased as the number of previously set kernels increased (Eq. [7]). This approach is consistent with Freier et al. (1984), who observed that the success of later-formed kernels decreased as the number of prior pollinations on the same ear increased. After taking within-ear synchrony into account, the model predicted kernel numbers per hectare accurately for P3978 at all pollen shed densities (Fig. 8b). The modified procedure, however, still overestimated kernels per hectare for P3925 by [approximately equal to] 10%.

A number of studies have reported a delay in subapical ear development relative to the apical ear as plant population density increases (e.g., Otegui, 1997). In this study, subapical ears produced kernels on a number of P3925 plants (1.2 ears per plant on average), but none formed on P3978 plants. Because we did not measure silk exsertion on these subapical ears, it was necessary to assume that their silk numbers and the timing of exsertion were identical to that of the apical ears. Measurements of silk exsertion on subapical ears on a similar prolific hybrid (Asgrow 740) grown at several plant densities (Fig. 1) indicated this assumption likely caused us to overestimate the contribution of the subapical ears to kernel set in P3925. When silk exsertion from the subapical ears of P3925 followed the same relative pattern as for Asgrow 740, predicted kernel numbers for P3925 decreased to within 5% of measured values for the three lowest MF treatments (Fig. 8b).

It is important to emphasize the uncertainty associated with incorporating the potential impact of pollination asynchrony and delayed development of subapical ears. These functions likely are affected by population density and by the inherent prolificacy of each hybrid (Otegui, 1997; Carcova et al., 2000). Yet our results provide evidence that asynchrony between ears and within ears must be taken into account for accurate prediction of kernel numbers. This is particularly important when pollen amount limits kernel number.

Many authors have shown that kernel number per plant depends on IPAR around silking (Andrade et al., 2000; Lizaso et al., 2001). Such results support the view that the supply of carbohydrates around silking imposes the primary limitation on the number of fertilized ovules that ultimately develops into kernels. There are numerous field conditions such as drought (Hall et al., 1982; Edmeades et al., 1993), flooding (Lizaso and Ritchie, 1997), or hybrid seed production in which asynchrony of male and female flowering could impose a limit on pollen availability to fertilize exposed silks (Bassetti and Westgate, 1994). Kernel set under such conditions would be constrained by the number of fertilized ovaries, even though concurrent metabolic responses to stress also contribute to decreased kernel set. The procedure presented herein predicts kernel set based primarily on the synchrony of staminate and pistillate flowering, delayed development of subapical ears, and the detrimental effect of prior fertilization on the success of later-fertilized ovaries. A general procedure to forecast kernel numbers under both assimilate-limited (Lizaso et al., 2001) and pollen-limited conditions is forthcoming.

Abbreviations: IPAR, intercepted photosynthetically active radiation; MF, male fertile; MS male sterile; [P.sub.ind], population index; RMSE, root mean square of error.
Table 1. Parameters used in Eq. [4] to predict daily silk exsertion
for subapical ears of Pioneer 3925 (P3925). Silk exsertion curves
were fit to measured field data for the apical ear of P3925 and for
apical and subapical ears of Asgrow 740. Parameters for subapical
ears of P3925 were calculated relative to the apical ear using the
ratio of apical/subapical values for the corresponding Asgrow 740
parameters. Subapical ear parameters for P3925 were adjusted to 5.7
plants [m.sup.-2] by linear interpolation.

                                        1 plant
Eq. [4] parameters                    [m.sup.-2]

                                      Asgrow 740

                         Apical ear                 Subapical ear

S[N.sub.x] ([dagger])
  (max silk no.)            808                         808
b ([double dagger]),
  [d.sup.-1]                  0.35                        0.35
[t.sub.0] ([section]),
  day of year               195.1                       195.8
duration, d                  10                          10

                                     Pioneer 3925

                                      Apical ear

S[N.sub.x] (max silk
  no.)                                  621
b, [d.sup.-1]                             0.486
[t.sub.0], day of year                  199.81 n
duration, d                               8

                                        8 plants
Eq. [4] parameters                     [m.sup.-2]

                                       Asgrow 740

                         Apical ear                   Subapical ear

S[N.sub.x] ([dagger])
  (max silk no.)           642.44                         642.44
b ([double dagger]),
  [d.sup.-1]                 0.46                           0.21
[t.sub.0] ([section]),
  day of year              196.4                          197.1
duration, d                  9                             12

                                      Pioneer 3925

                                      Subapical ear

S[N.sub.x] (max silk
  no.)                                   600.51
b, [d.sup.-1]                              0.329
[t.sub.0], day of year                   200.39
duration, d                                8.8

([dagger]) S[N.sub.x], the potential number of silks per ear equal to
the potential number of florets.

([double dagger]) b, shape parameter controlling the slope of the
curve.

([section]) [t.sub.0], time when the first silk is exserted.

Table 2. Observed and predicted number of florets and kernels per
unit land area produced at various levels of male fertility (MF)
for Pioneer 3978 and 3925 (P3978 and P3925). Data in parenthesis
indicate the floret or kernel value as a percentage of the observed
number of florets in each MF treatment. Difference (Diff) = Predicted
% - Observed %. Predicted kernel numbers are not corrected for loss
due to pollen viability, within-ear pollination asynchrony, or delay
in silk exsertion on subapical ears.

                                        Number of florets

Treatment                     Observed            Predicted       Diff

                                  florets [ha.sup.-1]              %

                                              P3978

100% MF ([dagger])        4.51 x [10.sup.7]   4.51 x [10.sup.7]    0
                                (100)               (100)
50% MF ([double           4.60 x [10.sup.7]   4.60 x [10.sup.7]    0
  dagger])                      (100)               (100)
20% MF ([section])        4.81 x [10.sup.7]   4.80 x [10.sup.7]    0
                                (100)               (100)
0% MF ([paragraph])       4.67 x [10.sup.7]   4.66 x [10.sup.7]    0
                                (100)               (100)

                                              P3925

75% MF (#)                3.38 x [10.sup.7]   3.37 x [10.sup.7]    0
                                (100)               (100)
50% MF ([dagger]          3.15 x [10.sup.7]   3.14 x [10.sup.7]    0
  [dagger])                     (100)               (100)
20% MF ([double dagger]   3.60 x [10.sup.7]   3.59 x [10.sup.7]    0
  [double dagger])              (100)               (100)
0% MF ([section]          3.82 x [10.sup.7]   3.81 x [10.sup.7]    0
  [section])                    (100)               (100)

                                         Number of kernels

Treatment                     Observed            Predicted       Diff

                               kernels             kernels
                             [ha.sup.-1]         [ha.sup.-1]        %

                                               P3978

100% MF ([dagger])        3.77 x [10.sup.7]   4.35 x [10.sup.7]   +12.9
                               (83.6)              (96.5)
50% MF ([double           3.78 x [10.sup.7]   4.34 x [10.sup.7]   +12.2
  dagger])                     (82.2)              (94.4)
20% MF ([section])        3.65 x [10.sup.7]   4.35 x [10.sup.7]   +14.7
                               (75.9)              (90.6)
0% MF ([paragraph])       1.90 x [10.sup.7]   2.40 x [10.sup.7]   +10.8
                               (40.7)              (51.5)

                                               P3925

75% MF (#)                2.22 x [10.sup.7]   2.89 x [10.sup.7]   +20.1
                               (65.7)              (85.8)
50% MF ([dagger]          2.05 x [10.sup.7]   2.45 x [10.sup.7]   +12.9
  [dagger])                    (65.1)              (78.0)
20% MF ([double dagger]   1.96 x [10.sup.7]   2.55 x [10.sup.7]   +16.6
  [double dagger])             (54.4)              (71.0)
0% MF ([section]          0.43 x [10.sup.7]   0.42 x [10.sup.7]    -0.3
  [section])                    -11.3              (11.0)

([double]) 70676 plants [ha.sup.-1]; 0.97 ears per plant.

([double dagger]) 71395 plants [ha.sup.-1]; 0.98 ears per plant.

([section]) 73791 plants [ha.sup.-1]; 0.99 ears per plant.

([paragraph]) 70916 plants [ha.sup.-1]; 1.00 ears per plant.

(#) 52019 plants [ha.sup.-1]; 1.07 ears per plant.

([dagger][dagger]) 44742 plants [ha.sup.-1]; 1.16 ears per plant.

([double dagger][double dagger]) 54445 plants [ha.sup.-1]; 1.09
ears per plant.

([section][section]) 54714 plants [ha.sup.-1]; 1.15 ears per plant.

Table 3. Predicted kernel number at decreasing levels of pollen
viability for the 20% male fertile (MF) treatment for Pioneer
3978 (P3978) and the 50% MF treatment for Pioneer 3925
(P3925). Kernel set was limited by pollen amount in both
treatments. The critical pollen rate needed to achieve 96%
kernel set in Eq. [6] was increased mathematically to simulate
loss of pollen viability below the original level assumed from
Bassetti and Westgate (1994). Predicted kernel numbers are
presented as a fraction of the observed values. Note that a
37% decrease in relative pollen viability resulted only in a 3
to 5% decrease in kernel set.

                                           Relative
                                         kernel number
                                           predicted

                       Relative pollen
Critical pollen rate      viability      P3978   P3925

grains [cm.sup.-2]
  [d.sup.-1]
100                         1.00          1.19    1.18
110                         0.91          1.18    1.17
120                         0.83          1.17    1.17
130                         0.77          1.16    1.16
140                         0.71          1.15    1.15
150                         0.67          1.14    1.15
160                         0.63          1.13    1.14


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J. I. Lizaso, M. E. Westgate, * W. D. Batchelor, and A. Fonseca

J.I. Lizaso and W.D. Batchelor, Dep. of Agricultural and Biosystems Engineering; M.E. Westgate and A. Fonseca, Dep. of Agronomy; Iowa State Univ., Ames, IA 50011. Received 15 May 2002. * Corresponding author (westgate@iastate.edu).
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