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Published online 19 March 2008
Published in Crop Sci 48:408-416 (2008)
© 2008 Crop Science Society of America
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Seed Size Variation in Grain Crops: Allometric Relationships between Rate and Duration of Seed Growth

V. O. Sadrasa,* and D. B. Eglib

a South Australian Research and Development Institute and Univ. of Adelaide, Waite Campus, Adelaide, Australia
b Dep. of Plant and Soil Sciences, Univ. of Kentucky, Lexington, KY 40546-0312

* Corresponding author (sadras.victor{at}saugov.sa.gov.au).


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Genetic control and environmental modulation of seed size operate through their influences on rate and duration of seed filling, and their interaction. To account for this interaction, here we advance an allometric model centered on the scaling exponent {alpha} calculated as the slope of the linear regression between duration and rate of grain filling in a log-log scale. The scaling exponent allows for three types of responses: seed size is stable as a result of full compensation between rate and duration ({alpha} = –1), seed size is variable as a result of rate ({alpha} > –1), or duration-dominated growth ({alpha} < –1). The concept was tested with 45 data sets from the literature involving nine crop species, and sources of variation including genotype, environment, and their interaction. Relative variation in seed size ranged from 5 to 274%, and the scaling exponent was strongly concentrated in the range from 0 (large, rate-driven seed size range) to –1 (narrow seed size range due to mutually cancelled effects of rate and duration). The range of seed size declined when the scaling exponent declined from approximately 0 to –1. An {alpha} {approx} –1 (rate and duration effects cancel each other) is necessary and sufficient for small variation in seed size, whereas {alpha} {approx} 0 is necessary but not sufficient for large seed size variation. The magnitude of seed size variation is dependent on the variation in the rate of seed growth when {alpha} {approx} 0. This double condition for seed size variability is summarized in a multiple regression model with {alpha} and range of rate of grain filling as independent variables, which accounted for 73% of the variation in range of seed size.

Abbreviations: GxE, genotype x environment • LS, least squares • RMA, reduced major axis


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
GRAIN YIELD IS DETERMINED by the number of seeds produced by the crop and their average size (weight per seed). Seed number is often more variable and more closely associated with yield than seed size (Egli, 1998; Peltonen-Sainio et al., 2007). In both domesticated and wild species intraspecific variability in seed size is small compared to the variability in seed number (Bradshaw, 1965; Harper et al., 1970). Recently, it has been proposed that this relatively narrow range of seed size in grain crops is a primary result of stabilizing natural selection (Sadras, 2007). In some species, including wheat (Triticum spp.) and grain legumes, agronomic selection may have reinforced natural selection for relatively narrow seed size, whereas the reduction in plasticity in seed number in cultivated sunflower (Helianthus annuus L.) and maize (Zea mays L.) may have increased seed size plasticity in comparison to their wild ancestors (Sadras, 2007). (Phenotypic plasticity is "the amount by which the expressions of individual characteristics of a genotype are changed by different environments" [Bradshaw, 1965:149]. In this paper, plasticity is used in a broader sense, as equivalent to variability, defined in turn as "the potential or propensity to vary" [Wagner and Altenberg, 1996:969]). A clear understanding of these relationships and their effect on yield may hinge on our knowledge of the fundamental characteristics of seed size, as driven by the rate and duration of growth and their interaction.

Interestingly, the physiological viewpoint of Egli (2006) and the evolutionary perspective of Sadras (2007) converged to identify a key role of seed size in the determination of seed number. From the perspective of compensation of yield components, Egli (2006) pointed out that seed number often adjusts in response to genetic variation in seed size. Timing of key events is important in elucidating physiological cause–effect relationships between size and number. Whereas actual seed size is primarily determined after grain set, potential seed size is defined before anthesis, during carpel formation (Calderini et al., 1999a, 1999b). In a simplified version of previous models (Charles-Edwards et al., 1986), Sadras (2007) shifted the focus from physiology to evolution, and formalized the hypothesis that an amount of resources R is used to produce a certain number of seed (SN) as a function of a uniform, environment-dependent target seed size (k) which maximizes the fitness of the mother plant (i.e., SN = R/k). There is evidence therefore to suggest that seed size plays an important role in modulating the genetic and environmental control of seed number (Egli, 2006; Sadras, 2007).

Egli (2006) summarized our understanding of the physiology of seed growth. He analyzed maximum seed size (A) in terms of average rate and duration of the filling period:

Formula 1[1]
Briefly, Egli (2006) emphasized the genetic basis of seed growth rate accounting for both inter- and intraspecific variation; for example, typical rates ranging from 1.3 mg seed–1 d–1 in rice (Oryza sativa L.) to 20 mg seed–1 d–1 or more in grain legumes. Characteristic seed filling durations in the range from 20 to 30 d, when based on the effective filling period, are frequently constrained by temperature or water availability in agriculturally relevant environments. Plant factors, including source–sink ratios, and environmental factors including temperature, water, and N availability, often have differential effects on rate and duration of grain filling, thus leading to variable seed size response. The aim of this paper is to characterize the intraspecific variation of seed size using a novel allometric approach to account for the integrated effect of rate and duration of grain filling.

ALLOMETRIC ANALYSIS
Allometric analysis is a particular case of scaling analysis dealing with the relative sizes of the organs of plants and animals (e.g., leaf vs. root, liver vs. heart) or process (e.g., body size vs. metabolic rate) (Coleman et al., 1994; Makarieva et al., 2003; McConnaughay and Coleman, 1999; Niklas, 1994; Ohnmeiss and Baldwin, 1994; Sadras and Wilson, 1997a, 1997b; Sládek et al., 2006; Withers and Hillman, 2001). The pioneering work of Pearsall (1927) established an allometric relationship between stem and root mass at time t:

Formula 2[2]
where β expresses the relative initial sizes of stem and root, and {alpha} is the ratio of the logarithmic growth rates. In the scaling literature, β is termed the scaling coefficient and {alpha} is the scaling exponent (Niklas, 1994).

By analogy with the allometric model of Pearsall, Sadras et al. (2007) proposed a generalized scaling expression of Eq. [1]:

Formula 3[3]
This model allows for the quantitative test of three alternative responses of seed size to genetic and environmental drivers: seed size is stable as a result of full compensation between rate and duration ({alpha} = –1) or seed size is variable as a result of rate ({alpha} > –1) or duration-dominated growth ({alpha} < –1).

Hypotheses
Our working hypotheses are that (i) {alpha} close to zero is required for large intraspecific variation in seed size; this is an expectation from the established dominance of rate of grain filling as the major source of intraspecific variation in seed size (Egli, 2006), and (ii) small variation in seed size, resulting from compensating effects of rate and duration of seed filling, are reflected in {alpha} close to –1. Graphically, a plot of range of seed size against {alpha} should show a declining trend for {alpha} between 0 and –1.


   METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Data Sources
Table 1 summarizes the data sources used for the analysis of variation in seed size. Most experiments were performed under field conditions, and the rates and durations of seed growth were derived from either (i) fitted curves (variants of bilinear, polynomial, or logistic models), (ii) estimated rate and final seed size (Daynard et al., 1971), or (iii) estimated rates and phenological observations (e.g., Bagnara and Daynard, 1982). In experiments such as that by Sexton et al. (1994) where a set of common bean (Phaseolus vulgaris L.) cultivars was grown at two locations, the analysis included each site in isolation, to derive a measure of genotype effect (G), and the combined data set to produce a measure of the interaction between genotype and environment (GxE). In this paper, the main focus is intraspecific variation (G). Variation in seed size driven by environmental factors, excluding temperature, experimental manipulation (e.g., grain removal), and GxE, was considered for comparative purposes.


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  		Table 1. Sources of data for the allometric analysis of rate and duration of seed filling. Rates and durations were calculated from fitted curves (Method FC), from final size and estimated rate (Method SR), or from estimated rate and duration based on phenological observations (Method RP).

 
Statistical Approach
Niklas (1994) and Coleman and McConnaughay (1995) discussed the statistical approaches to calculate the scaling exponent {alpha}. For predictive purposes, Model I regression (least squares [LS]) produces appropriate estimates of the scaling exponent {alpha}LS. However, to account for the fact that both variates in the model (i.e., rate and duration) are measured with error, Model II regression (reduced major axis) could be more appropriate ({alpha}RMA). The scaling coefficients are related according to {alpha}RMA = {alpha}LS/|r|, where r is the coefficient of correlation. This means that the stability of {alpha}RMA will decrease when the relationship between variates is weaker. A secondary aim of this paper was thus to compare {alpha}RMA and {alpha}LS.

Some papers reported rates and durations using time (e.g., mg seed–1 d–1, d) and others used thermal time (e.g., mg seed–1 per degree day [°Cd–1], °Cd). The scaling exponent {alpha} used as a reference is, however, independent of the units of the variates from which it is derived (Niklas, 1994; Reiss, 1986; Xiao, 1998) provided rates and durations are expressed on the same basis, and temperature is not a major source of variation. Ranges of seed size, rate and duration of grain filling were calculated as 100 x (maximum – minimum)/minimum.


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Comparison of {alpha}RMA and {alpha}LS
By definition (Niklas, 1994), {alpha}RMA and {alpha}LS yield the same information when relationships are tight (r close to 1), but {alpha}RMA increases relative to {alpha}LS when relationships are weaker. This was reflected in the scatter plot of {alpha}RMA vs. {alpha}LS (Fig. 1 ) where points were distributed around the y = x line, but with a much larger range for {alpha}RMA (from –5.53 to 1.13), compared to {alpha}LS (from –1.88 to 0.51). From statistical and biological considerations mostly concerned with interspecific comparisons, Niklas (1994) concluded that neither Model I or Model II methods have universal application, and that the choice of methods depends on the nature of the data. For the following analyses in this study, we used least squares on the grounds of more stable scaling coefficients, and provided further comparisons between {alpha}RMA and {alpha}LS (e.g., Fig. 2 ).


Figure 1
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  		Figure 1. Comparison of scaling exponents calculated with least squares ({alpha}LS) and reduced major axis regression ({alpha}RMA).

 

Figure 2
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  		Figure 2. Examples of intraspecific scaling relationships between rate and duration of seed growth. Multiple symbols for a cultivar indicate different experiments or seasons, except for sorghum where closed and open symbols indicate apical and basal grains, respectively. The solid line is the least squares regression, and dashed lines are isolines of seed size with {alpha} = –1. Standard errors (SE) of the scaling exponents are also shown. Data sources: maize, Echarte et al. (2006); sunflower, López Pereira et al. (1999); rice, Fujita et al. (1984); soybean (control treatment), Egli (1999); sorghum, Gambin and Borrás (2007). For rice and soybean, rate is in mg seed–1 d–1 and duration in d, and for maize, sunflower, and sorghum rate is in mg seed–1 per degree day (°Cd) and duration in °Cd. Variate units do not affect the magnitude of the scaling exponent.

 
Allometric Analysis of Rate and Duration of Seed Growth
Figure 2 illustrates the allometric relationship between rate and duration of seed growth for contrasting case studies. These particular experiments with soybean [Glycine max (L.) Merrill] and sorghum [Sorghum bicolor (L.) Moench] showed relatively stable seed size, and scaling exponents correspondingly close to –1. In contrast, the scaling exponents for sunflower and maize were significantly greater than –1 (P < 0.05), with very flat lines reflecting a rate-dominated seed growth. In a few cases, scaling exponents were positive, as illustrated for a rice data set (Fig. 2). In no case, however, did positive scaling exponents reach statistical significance (P < 0.05). The application of an allometric model to particular case studies therefore allows for a quantitative characterization of seed growth in terms of the relative contribution of rate and duration of seed growth to seed size variation. Furthermore, this approach allows for additional insight into key interactions such as genotype x season or genotype x grain position.

For instance, the relative stability of the scaling exponent across seasons for soybean, sunflower, and maize (Fig. 2) can be taken as a measure of a small genotype x season interaction. Whereas standard analysis of variance for seed size could give an indication of this interaction, the allometric approach adds to the interpretation of this important source of variation. In contrast to the seasonal-stable scaling exponents for sunflower and maize depicted in Fig. 2, the experiments of Swank et al. (1987) with soybean showed a shift in {alpha}LS from –1.89 in 1982, to –0.10 in 1983 (points in Fig. 3 inscribed in a circle or square, respectively). In both years, variation in seed size was large (i.e., 193% in 1982, and 130% in 1983). Analysis of variance could have shown a slight effect of genotype x season on seed size, whereas allometric analysis of rates and duration revealed a shift from duration-driven differences in 1982 to rate-driven in 1983. This interaction is particularly interesting, because the genotypes in this study were carefully selected from a larger group of genotypes, using data from 1982, for large differences in seed size and duration of seed filling with minimal variation in seed growth rate. Departures from these conditions in 1983 due to seasonal interactions were captured by the allometric analysis.


Figure 3
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  		Figure 3. Relationship between seed size range and {alpha}LS, the scaling exponent relating duration and rate of seed growth derived from the data sets summarized in Table 1. Emphasis is on (a) crop species and (b) sources of variation. The points inscribed in a circle or a square indicate data from the experiment of Swank et al. (1987) where cultivars were intentionally selected for comparable rates and large variation in duration of seed growth. Sources of variation where attributed to genotype (G) when experiments included several cultivars in a single treatment, site or season; environment (E) when a single genotype was grown under a range of conditions (e.g., sites, seasons, water regime), or GxE when both cultivars and environments were combined.

 
Analysis of the data of Gambin and Borrás (2005) with sorghum (Fig. 2) showed the scaling exponents were similar for apical ({alpha}LS = –0.50) and basal grains ({alpha}LS = –0.42), indicative of a small genotype x grain position interaction. In contrast, analysis of data reported by Egli et al. (1978) for soybean revealed scaling exponents indicative of rate-dominated variation in seed size for early pods ({alpha}LS = –0.06) that shifted to a more balanced rate–duration relationship, and more stable seed size, for late pods ({alpha}LS = –0.77). This change in the scaling exponent corresponded with the reduction in seed size range from 160 to 84%.

Relationship between Seed Size Range and the Scaling Exponent {alpha}
Figure 3 shows the relationship between the range of seed size and the scaling exponent {alpha}LS for the pooled data; the relationship with {alpha}RMA was much looser (not shown). Variation in seed size ranged from negligible (5%) to 274%, and the scaling exponent was strongly concentrated in the range from 0 (large, rate-driven seed size range) to –1 (narrow seed size range, rate, and duration effects mutually cancelled). A data point from the experiment of Swank et al. (1987) where soybean cultivars were selected for comparable rates and large variation in duration of seed growth, had the lowest {alpha}LS (–1.89) indicative of duration-driven variation in seed size, and a correspondingly high seed size range (encircled point in Fig. 3a).

As expected from first principles (see Hypotheses presented earlier), the range of seed size declined when {alpha}LS declined in the approximate range from 0 to –1 (Fig. 3). An important distinction in this relationship is that {alpha} {approx}–1 is necessary and sufficient for small variation in seed size, whereas {alpha} {approx} 0 is necessary but not sufficient for large seed size variation, hence the scatter of data in the vicinity of {alpha} {approx} 0. Indeed, irrespective of the actual magnitude of rate and duration of seed filling, {alpha} {approx} –1 means that the variation in seed size range due to rate and the variation due to duration cancel each other, as illustrated for the soybean and sorghum data sets in Fig. 2. In contrast, {alpha} {approx} 0 is necessary for rate-driven variation in seed size but the magnitude of seed size variation is also dependent on the actual magnitude of the variation in the rate of seed growth. This is illustrated in the comparison of the data sets of maize and sunflower in Fig. 2, where the fitted lines account for the effect of GxE. For both data sets, the scaling coefficients are comparably close to zero, that is, {alpha}LS = –0.16 ± 0.066 for sunflower and {alpha}LS = –0.30 ± 0.127 for maize. The actual variation in seed size range is, however, 183% for sunflower and 28% for maize. This difference is accounted for by the range of variation in rate of grain filling, that is, 250% in sunflower and 30% in maize. For the pooled data set, excluding the point with duration-dominated seed size variation (encircled point in Fig. 3a), the dual condition of a large (close to zero) scaling exponent and a large range of rate of seed filling (R%) required for large variation in seed size is summarized in the model:

Formula 4[4]
All three parameters in Eq. [4] were significant at P < 0.0001, and there was no association (P = 0.19) between the scaling exponent {alpha}LS and the range of seed filling rate R%. To allow for a two-dimensional graphic depiction of the relationship implicit in Eq. [4], we plotted seed size range against range of seed filling rate (Fig. 4a ) and the residuals of this regression against the scaling exponent (Fig. 4b). Although the range of the rate of seed filling accounted for a large proportion of the variation in the range of seed size (r2 = 0.59), the relationship was quite scattered. For instance, for a range of seed filling rate around 150%, seed size range varied between 38 and 159% (Fig. 4a). This variation was partially related to the interplay between rate and duration, which was in turn captured by the scaling exponent (Fig. 4b).


Figure 4
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  		Figure 4. Relationships between (a) seed size range and range of rate of seed filling, and (b) the residuals of the relationship in (a) and the scaling exponent {alpha}LS.

 
The pooled data in Fig. 3 and 4 indicated that variation in rate of seed growth is by far the most common contributor to seed size variation, while the compensation ({alpha}LS = –1.0) that minimizes seed size variation is rare. Most of the genotypes in this analysis are improved cultivars, selected for high productivity (yield, lodging, and disease resistance, etc.) and adaptation to particular environments. Both natural and agronomic selection for adaptation (maturity) in combination with temperature and water restrictions on crop duration might have limited the variation in seed fill duration and encouraged domination by rate. Selection for yield was probably irrelevant, given that rate-dominated variation in size is yield neutral (Egli, 1998, 2006).

Interspecific Comparisons of Seed Size Range and the Scaling Exponent {alpha}
Comparison among species was constrained to wheat (n = 49), soybean (n = 28), and maize (n = 20), the three crops with sufficient data for more detailed analysis. For these species, the measure of relative seed size plasticity reported by Sadras (2007) aligned with the average scaling exponent (Fig. 5 ). The larger, closer to zero scaling exponent for maize (average {alpha}LS = –0.06) was consistent with the relatively high seed size plasticity in this species, which in turn was attributed to domestication which led to strong apical dominance, reduced prolificacy and hence restricted seed number plasticity (Sadras, 2007). Bradshaw's (1965) concept of a hierarchy in plasticity, whereby the stability of a given trait (e.g., seed size) can be considered at least partially the outcome of the plasticity in other traits (e.g., seed number) is at the core of this interpretation. In comparison, species such as soybean (average {alpha}LS = –0.40) and wheat (average {alpha}LS = –0.34), where domestication did not seriously compromise prolificacy, plasticity of seed number remained high, and seed size plasticity relatively low (Sadras, 2007).


Figure 5
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  		Figure 5. Relationship between seed size plasticity and the scaling exponent {alpha}LS for maize, wheat, and soybean. Both variables are dimensionless. Seed size plasticity is the variability in seed size relative to the variability in seed number derived from Sadras (2007) (his Fig. 4b), and {alpha}LS is the average for each species from Fig. 3.

 
Limited evidence indicates that rice might depart from the hypothetical association in Fig. 5, that is, it is a highly prolific species with large capacity to accommodate resource variation through grain number (Hayashi et al., 2007; Kumar et al., 2006; Saito et al., 2006), yet there were few but strong cases of rate-driven, seed size plasticity comparable to sunflower and maize (Fig. 2, 3). A relatively large genetic diversity in this species may contribute to the larger than expected range of seed size; indeed the experiment of Fujita et al. (1984) showing {alpha}LS = 0.28 and seed size range = 274% (Fig. 2 and 3) involved a two- to sixfold difference in seed size between javanica and indica types, with much narrower ranges within groups. In comparison to other cereals such as wheat and barley (Hordeum vulgare L.) that are consumed after milling or other transformations, the integrity, shape, and size of the rice grain are critical marketing elements and major breeding targets (Rabiei et al., 2004; Shi et al., 2000).

Our allometric analysis therefore complements evolutionary and agronomic considerations accounting for the trade-off between seed size and number (Sadras, 2007), and suggests that the scaling exponent can contribute a physiological explanation to the variable plasticity in seed size of crop species. Interestingly, comparative studies indicated that seed structure and composition and plant characteristics (e.g., C3 vs. C4) had no systematic influence on rate and duration of seed growth (Egli, 1981, 1998, 2006; Egli and Bruening, 2007). Thus, while recognizing the limited span and the degree of speculation in the interpretation of the relationship in Fig. 5, we suggest that the allometric integration of rate and duration might shed new light where previous analyses of rates and durations in isolation did not reveal species-specific patterns.


   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
Allometric analysis allowed for an original, integrated perspective on the interplay between rate and duration of seed filling, which in turn accounts for the genetic and environmental factors modulating seed size in grain crops. Independently of the magnitude of the variation in rate and duration, {alpha} {approx} –1 corresponds with little variation in seed size, as it accounts for the mutual cancellation of rate and duration effects. Large, rate-driven variation in seed size requires both {alpha} {approx} 0 and large variation in rate. This allometric approach could be useful for evolutionary, agronomic, and physiological analysis of seed size, and may also be used for other processes such as leaf growth or accumulation of anthocyanins in fruits, where a framework of rates and durations is applicable (Sadras et al., 2007).


   ACKNOWLEDGMENTS
 
We thank G.A. Slafer for discussion of temperature effects on the rate and duration of seed growth. Work by VOS is partially funded by the River Murray Improvement Program and DBE was funded by the Kentucky Agricultural Experiment Station.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
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Received for publication May 23, 2007.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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