International Journal of Agronomy

Volume 2018, Article ID 9658707, 7 pages

https://doi.org/10.1155/2018/9658707

## The Ideotype for Seed Size: A Model Examining the Relationship between Seed Size and Actual Yield in Pea

Department of Plant Sciences and Plant Pathology, Montana State University, Bozeman, MT 59717, USA

Correspondence should be addressed to Norman F. Weeden; ude.anatnom@nedeewn

Received 23 December 2017; Accepted 14 March 2018; Published 1 August 2018

Academic Editor: Othmane Merah

Copyright © 2018 Jamin Smitchger and Norman F. Weeden. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Seed size plays a large role in determining productivity of large seeded legumes. In many large seeded legumes such as pea and bean, actual yield, defined here as grain yield at harvest minus the weight of seed planted, is often a better measure of actual productivity than grain yield at harvest, because the weight of planted seed varies with seed size. In many grain legumes, the weight of planted seed can be equal to 10% of the total grain yield, and minimizing the weight of planted seed could significantly impact actual yield. This study produced an algorithm to examine the relationship between seed size, yield, and actual yield in silico. The output of this algorithm predicted the ideotype for seed size in peas to be much lower (12.5 g./100 seeds) than the seed size of nearly all commercial varieties, indicating that efficiency in pea cropping systems could be increased by reducing seed size. Modifications to the algorithm would allow the prediction of the ideal seed size in other legumes. The algorithm predicts that there is likely very little correlation between seed size and grain yield, although larger seeded legumes will likely have a higher harvest index. Plant breeders can use the ideotype predicted by this function to create varieties of peas and other large seeded legumes that have higher actual yield. The ideotype for seed size was defined in pea.

#### 1. Introduction

In dryland production regions, pea yield is mainly limited by summer heat and availability of soil water. Although pea has poor yield stability in dryland cropping systems [1], future yield potential can be estimated based on prior yields, allowing the breeding of advanced cultivars that can thrive in a specific region. Averaged across 2001-2016, the yield potential in Montana, the largest pea production region in the US, is 1735 kg/ha [2]. Because environmental conditions are the limiting factor, maximizing economic returns within the constraints of a specific environment is a major breeding objective.

Actual yield as defined here is the weight of seed harvested minus the weight of seed planted. Particularly in large seeded legumes in low yielding environments, yield* per se* is not a perfect measurement of economic productivity. Given two varieties of peas with the same yield potential (1735 kg/ha) but different seed sizes, where variety A has a 100 seed weight of 15 g and variety B has a 100 seed weight of 30 g, variety A would have 124 kg/ha higher actual yield than variety B at current seeding rates of 80 seeds/m^{2}. Actual yield is a better measure of productivity than the common measurement of grain yield at harvest because it accounts for seed cost. Given seeding rates for pea of 170 kg/ha and an average crop yield of 1735 kg/ha), the ratio between harvested grain and seeds planted (harvest ratio) is approximately 10:1. Based on current seeding rates and yields, actual crop yield declines 0.46% with every 1 g increase in hundred seed weight. Chickpeas, beans, and other large seeded legumes also have a similar harvest ratio, especially in many dryland cropping systems. This ratio is low compared to most other crops. Camelina and canola, for example, have a harvest ratio of ~500:1, and wheat has a harvest ratio of ~40:1 (assuming a seeding rate of 78.5 kg/ha and 50-bushel grain yield).

Yield potential is the genetically determined ability of a crop to generate optimal yield in a given growth environment [3]. Yield potential is thought to be partially determined by seed size, and numerous studies have tried to understand the relationship between seed size and yield in pea and other pulses with contradictory results [1, 4–9]. A lack of correlation between seed size and yield was found in grass pea under drought conditions [5]. In chickpea, a positive correlation was found between seed size and yield, but no effect was seen in lentil [4]. In one study in pea, a strong correlation was found between seed size and yield [6], and Krajewski et al. [8] found that seed size was positively correlated with yield, explaining 20% of the variation in yield in pea. However, peduncle number explained nearly 50% of the variation in yield, indicating that seed size is not the most important factor in determining seed yield. Other research has also shown a correlation between the number of fertile nodes and seed yield rather than seed weight and seed yield [10]. Timmerman-Vaughn et al. [1] indicated that seed size is negatively correlated with seed yield, contradicting the study by Krajewski et al. [8], and Irzykowska and Wolko [7] indicated that QTLs for seed size and yield were not colocated. Gambin and Borras [11] concluded that crop yield within a species was more related to variations in seed number than in seed weight. Hence, there appears to be no consensus regarding the relationship, and the ideotype as defined by Donald [12] for seed size in pea has not been defined.

The theoretical effect of seed size on grain yield is positive because larger seeded peas have a higher harvest index than smaller seeded peas (when all other factors are held constant). There may be an association between low pod wall proportion and high seed yield [13], but seed weight did not seem to affect yield in this study. Pate et al. [3] in a previous study in lupin estimated that for every 100 units of carbon imported from the parent plant into pods or seeds, 52 are incorporated into seeds, 37 are into the pod, and the remaining 11 units are lost as CO_{2}, indicating that up to 41.5% of photosynthate may end up in pod material. However, 96% of the nitrogen in the mature fruit is incorporated into seeds, with 16% of that being remobilized from pod material, indicating that seeds may be more nutrient rich than pods. Pate et al. [3] indicated that approximately 7% of carbon is photosynthesized from the pod itself. Flynn and Pate [14] indicated that approximately an eighth of the carbon in seeds (2 mg/ photoperiod after 10 days after anthesis) is synthesized from pods and that many of the nutrients in pod tissues are transported to the seed prior to senescence.

When compared to other legumes, peas have a much lower proportion of pod walls to percent pod biomass, and it is estimated that this percentage is around 13% [13]. A hypothetical plant that could produce 100 g of seed could produce 400 seeds that weighed a 0.25 g or 100 seeds that weighed 1 g [15]. However, in reality the amount of photosynthate required to create 1 kg of small seeded peas is higher than that required to create 1 kg of large seeded peas because small seeded peas require more pod surface area per unit of mass and more flowering nodes in order to produce the same amount of yield. Therefore, smaller seeded varieties may have an inherent theoretical disadvantage in achieving high grain yields. The consideration of the plasticity of yield components is important when determining the trade-offs between seed size and seed number [15]. Assuming that pod wall thickness is the same in large seeded and small seeded genotypes and that pod length is the same, we can determine the relative amount of photosynthate required to produce pods and seeds of a certain size. Pod material is photosynthetic so it may not be a major burden for the plant to produce, but pods are a sink during their formation [14]. Grain yield is associated with harvest index, which is the ratio of grain yield to plant biomass. This source-sink relationship in peas is dependent on seed size, pod length, plant height, leaf size, stem width, stem wall thickness, leaf thickness, stipule size, and other factors. Here we develop a mathematical model to examine the effect of pod length and seed size on the proportion of photosynthate available for partitioning into seed and pod material. All other parameters of harvest index were assumed to be fixed, and only pod size and seed size were varied.

#### 2. Materials and Methods

The functions to predict the ideal pod length and ideal seed size were created in Excel spreadsheets. All of the equations were derived from commonly used mathematical concepts.

##### 2.1. A Function to Predict the Ideal Pod Length

The formula for the surface area of a cylinder (SA=) allows the prediction of total pod surface area per plant when pod length is varied. Yield, seed size, spacing of seeds in the pod, and all other parameters were held constant. It is assumed that pea pods have the basic shape of a cylinder. Seed radius (r in the formula) is held constant and only pod length is varied. Thus, seed number is fixed, and the number of pods per plant will vary with pod length. For example, if a 4 cm pod holds 5 seeds and the yield potential is fixed at 100 seeds in the model, the plant will produce 20 pods. The surface area of the twenty pods based on the formula for the surface area of a cylinder is 188.4 cm^{2}. Changing the pod length to 8 cm will increase the number of seeds per pod, and only 10 pods will be needed to have the same yield potential. The surface area of the 10 pods is 172.7 cm^{2}. The pod surface area was compared between long-podded and short-podded pea to determine the ideotype for pod size.

##### 2.2. A Function to Predict the Ideal Seed Size

The core of the ideal seed size model is the equation for actual yield (yield per acre, seed planted). A number of equations (see below) are used to increase the theoretical accuracy of the model by predicting changes in harvest index based on seed size. The algorithm is based on the idea that increasing harvest index will result in increased yield, a trend which is commonly seen in other crops [16]. This algorithm did not account for variation in other factors besides seed size.

##### 2.3. Assumptions of the Function to Predict the Ideal Seed Size

(i)The density of a dry pea seed is 1.176 g/cm^{3}, which was empirically determined in this study from a single large sample of dry pea seed.(ii)The density of a dry pea pod is approximately 1.05g/cm^{3}, which was empirically determined from a single sample of pea pods.(iii)Seed weight (which also influences pod size and pod number when the yield potential is held constant) is the only characteristic of a plant that varies.(iv)The amount of photosynthate available to produce either pods or seeds is fixed at a specified yield potential dependent on specific environmental factors.(v)Formulas for the surface area and volume of a cylinder are a good approximation of the shape of a pea pod. This model does not account for the portion of the tip and base of the pod not occupied by seed.

Several other assumptions will be identified and discussed in the Results and Discussion.

The function to determine the ideal seed size uses the following equations: each one of the parameters is a column title in the Excel spreadsheet found in supplementary data and explains the underlying calculations occurring.

Seed planted per acre: ((seeding rate/m^{2})/(sq. feet per m^{2}))(square feet in an acre).

Seed cost multiplier (cost of seed planted per lb/ value of harvested grain per lb). This accounts for the difference in value between seed grown by the farmer and seed purchased by the farmer.

Mass per seed (Mass): 100 seed weight/100. This can be varied or held constant at a specific value in order to determine the ideal seed weight.

Volume of a seed: (Mass)/(seed density). Seed density is approximately 1.176 g/cm^{3}.

Radius of a seed (r): the radius of a seed is found by using the formula for the volume of a sphere (volume = 4/3r^{3}) and solving for the radius. Therefore r = .

Seed diameter (Seed Diam): 2 times the radius.

Pod length: this can be varied in the model or held constant at a specific value. Pod length is generally about 6 cm in peas, but this varies depending on the variety. This parameter affects seeds per pod, and pod number.

Correction for space between seeds in a pod: this accounts for the fact that there is often space between the seeds in a dry pod. The multiplier, which represents the actual space taken up by the seed divided by the dry seed diameter, accounts for the pod wall material required to produce a seed if the seed actually requires more space in the pod than its dry seed diameter. It should be noted that the multiplier increases pod material by decreasing the number of seeds per pod. Therefore, more pods must be produced if there is more space between seeds. Seed yield will likely decrease if more photosynthate is partitioned into pod material versus seed material.

Pod width: seed diameter + 2 x pod wall thickness. Pod wall thickness was determined empirically to be approximately 0.194 mm on average.

Photosynthate: this is the total amount of sugars and other nutrients available for partitioning into seeds, seed testa, and pods (in units of grams per plant). This parameter is held constant at an arbitrarily determined value. The algorithm assumes that pea plant sizes, leaf number, and other factors are the same. Only seed size and a few other yield parameters are varied. The percentage of photosynthate dedicated to seeds or pods will vary depending on seed size and pod number. It is quite often that pea plants have enough energy to create about 20 g of pod wall material, seeds, and seed testa, based on empirical data. Photosynthate (g) = (mass per seed((length of pod/ radius of seed)pod number)) + (pod number (h- r^{2}h)pod density in g/cm2). In this case h=pod length, r1 = pod width, and r = seed diameter. This equation can be partitioned into two components where (mass per seed((length of pod/ radius of seed)pod number)) is the seed yield component and (pod number (h- r^{2}h)pod density in g/cm2) is the pod weight component. The percentage of photosynthate being directed toward the seed or pods for any seed weight can be determined by dividing the seed yield component or pod weight component by the total weight of photosynthate.

Pod number based on amount of photosynthate produced (Pod #): this the number of pods per plant. It is important to note that, to produce the same amount of photosynthate per plant, pod number must vary when seed size varies. Pod number can be determined with the following formula. Pod # = photosynthate/ per seed(h /seed diameter))+((h- r^{2}h)pod . In this case h=pod length, r_{1}=half of the pod width, and r=seed radius. Photosynthate is held constant at an arbitrary value (10 g), and the weight of photosynthate is divided by the weight of a pod at a given seed size.

Amount of photosynthate allocated to seed: this is number of grams of photosynthate that is allocated to the seed and not allocated to the pods. The amount of photosynthate allocated to the seed is dependent on seed size and pod length. The equation for the amount of photosynthate allocated to seed = Mass(Pod L/Seed Diam)Pod#.

% Allocated to seed: (photosynthate allocated to seed/photosynthate)100.

Expected yield potential: this will vary depending on the environment and location. According to the National Agricultural Statistics Service the average yield in Montana during the 15-year period between 2001 and 2016 was 1735 kg/ha.

Expected weight of yield components: this is the total weight of pods, seed, and seed testa produced by a specific plant. The relative proportion of pods, seeds, and seed testa is dependent on seed diameter, and the total amount of pod material is required to achieve specific yield potential decreases with increases in seed size. All other factors are held constant. In general, the weight of seed is 85% of the weight of the total weight of all the yield components, so the expected yield in Montana (1735 kg/ha) is divided by .85. The expected weight of yield components is held constant in the seed size model, but the number is dependent on expected yield.

Yield at harvest: the expected yield at harvest of seed given a specific seed weight. Yield at harvest = (% Allocated to seed/100) Expected weight of yield components. Expected yield at harvest increases with increasing seed weight because less photosynthate is going to pod weight.

Actual seed yield: yield at harvest minus the weight of seed planted. This is essentially parabolic. It reaches a high point at the ideal seed weight and then starts to decline.

Volume of seed that is occupied by the testa: This is part of the volume of a seed that is composed of the seed coat. The seed coat (testa) is the outer surface of the seed, and it is often stripped from the seed during processing. It was empirically determined in this study that the seed testa has an average thickness of 0.0125 cm. The volume can be found by determining the volume of a seed and subtracting the volume of the seed without its testa. Volume of testa= 4/3- 4/3r^{3}. In this case r_{1} = seed radius, and r = seed radius minus testa thickness.

Mass of testa: volume of testaseed density. It is possible that the testa may be more or less dense than a whole seed, but it sinks in water, indicating that it has a density over 1 g/cm^{3}.

Actual seed yield minus testa: seed mass - mass of testa.

Number of g of seed: the number of grams of photosynthate dedicated to the seed without its testa.

Percent of photosynthate going to seed without its testa: (number of g of seed/photosynthate) 100.

Yield minus the weight of seed testa: expected weight of yield componentsPercent of photosynthate going to seed without its testa.

Weight of seeds planted with varying 100 seed weights: (total seeds per acreMass per seed) the number of grams per pound (there are 453.59237 g per lb).

#### 3. Results and Discussion

Our mathematical algorithm (results shown in Figure 1) was created to predict the ideotype for pod length. This function shows the relationship between pod surface area per plant and pod length.