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University of Guelph, Department of Plant Agriculture, Crop Science Bldg., Guelph, ON N1G 2W1, Canada
* Corresponding author (lizlee{at}uoguelph.ca).
| ABSTRACT |
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Abbreviations: DMA, dry matter accumulation G x E, genotype by environment interaction GFP, grain-filling period GIS, geographic information system HI, harvest index LAI, leaf area index PPFD, photosynthetic photon flux density RRS, reciprocal recurrent selection TPE, target population of environments
Received for publication April 10, 2007.
University of Guelph, Department of Plant Agriculture, Crop Science Bldg., Guelph, ON N1G 2W1, Canada
* Corresponding author (lizlee{at}uoguelph.ca).
During the maize (Zea mays L.) hybrid era (1939 to present), commercial grain yields have improved nearly sixfold and the genetic component of the improvement has been estimated as approximately 60%. In this paper, we examine physiological factors and successful breeding strategies that underlie the yield improvement. Grain yield is the product of accumulating dry matter and allocating a portion of the total dry matter to the grain. The processes influencing dry matter accumulation are commonly referred to as the "source" components, while the processes influencing allocation of dry matter to the grain are referred to as the "sink" components. On the source side, changes in leaf canopy size and architecture account for only a minor portion of the improvement. The majority of the improvement in source capacity is due to visual and functional "stay-green." On the sink side, the improvement is through changes in the relationship between kernel number per plant and plant growth rate during a period bracketing silking. In a breeding context, these improvements have been made (i) in a "closed" germplasm pool stratified into heterotic groups; (ii) through use of a pedigree method of breeding structured to mimic reciprocal recurrent selection and thereby improving both additive and nonadditive genetic effects; and (iii) by a gradual increase in plant population densities during the hybrid era as the constant source of stress during both inbred line development and hybrid commercialization. Functional stay-green and the sink establishment dynamics still represent opportunities for yield improvements. It is essential that source and sink are kept in balance, and that improvement in one accompanies a simultaneous improvement in the other. One strategy for exploiting these opportunities is to incorporate high plant population density trials into inbred line development programs.
Abbreviations: DMA, dry matter accumulation G x E, genotype by environment interaction GFP, grain-filling period GIS, geographic information system HI, harvest index LAI, leaf area index PPFD, photosynthetic photon flux density RRS, reciprocal recurrent selection TPE, target population of environments
| INTRODUCTION |
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Source–Sink Relationship
Maize grain yield can be dissected into component whole-crop level physiological processes that occur during various development phases in the life cycle of the plant (Tollenaar and Lee, 2006) (Fig. 2
). In short, grain yield is the product of accumulating dry matter (i.e., biomass) and allocating a portion of the total aboveground biomass to the grain. The processes influencing dry matter accumulation (DMA) are commonly referred to as the "source" components, while the processes influencing allocation of dry matter to the grain are referred to as the "sink" components. It is essential that source and sink are kept in balance, and that improvement in one accompanies a simultaneous improvement in the other. One way that abiotic stress acts on the maize plant is to shift source and sink processes out of balance with one another. Excess source capacity, relative to sink capacity, results in other tissues (e.g., leaves, stalks) acting as sinks. Purpling of leaves, sheath tissues, and stalks during the grain-filling period (GFP) are classic symptoms of excess source capacity (Fig. 3e and 3f
). Excess sink capacity, relative to source capacity, results in premature senescence of leaves and stalks during the GFP (Fig. 3g). In a modern short-season maize hybrid approximately 50% of the total seasonal dry matter is accumulated by flowering (i.e., silking and anthesis); with the remaining 50% of the seasonal dry matter being fixed during the GFP (Fig. 4
) (Tollenaar et al., 2004). In a mature maize plant, grown with conventional agronomic practices, approximately 50% of the total dry matter is allocated to the grain, with the stover (i.e., leaves, stalk, cob, husk tissues, and tassel) accounting for the remaining 50% (Fig. 4) (Tollenaar et al., 2004). This distribution of dry matter at maturity between stover and grain is referred to as the harvest index (HI). All of the dry matter allocated to the grain is fixed during the GFP; none of the dry matter in the grain is due to remobilization of dry matter which was fixed before flowering (Below et al., 1981; Cliquet et al., 1990).
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Commercial maize grain yield is not exclusively due to genetic improvement, as there have been substantial changes to agronomic practices during this 65- to 70-yr period as well. Starting somewhere in the 1960s commercial fertilizers were used, with increasing N levels occurring until the mid-1980s (cf., Troyer, 2004; Crosbie et al., 2006). Better weed control was achieved through use of herbicides (e.g., 2,4-D [2,4-dichlorophenoxy acetic acid] was first used commercially in 1945 and atrazine [2-chloro-4-ethylamino-6-isopropylamino-1,3,5-triazine] was first used commercially in 1965 [cf., Troyer, 2004]). More uniform distribution of plants within a field was achieved by reducing row widths from 102 cm to 76 cm in the mid-1960s to early 1970s (cf., Troyer, 2004). Earlier maize planting (Duvick, 1989; Kucharik, 2006) has effectively increased the duration of the growing season and, consequently, the period of time that plants can absorb incident solar radiation. Finally, plant population densities gradually increased from 30,000 plant ha–1 to 79,000 plants ha–1 (cf., Crosbie et al., 2006). In general, 60% of the increase in grain yield is attributed to genetic improvement (Duvick, 1992) with 40% being attributed to improved agronomic practices (Cardwell, 1982); however, realistically 100% of the increase in grain yield is actually due to the interaction between genetics and agronomic practices (Tollenaar and Lee, 2002). To make meaningful comparisons given the genetics x agronomics interaction, it is important to examine the genotype in the context of the agronomic management (i.e., plant population density) that was utilized at the time of release (Duvick et al., 2004).
To fully appreciate what physiological attributes have changed during the hybrid era, we should briefly examine what has not changed during the hybrid era. The magnitude of heterosis has not changed. While hybrid grain yields have increased, inbred line grain yields have mirrored them, resulting in no significant change in the magnitude of heterosis for grain yield (Duvick, 1999; Duvick et al., 2004). The maximum potential photosynthesis (i.e., photosynthesis at silking) has not changed (Tollenaar et al., 2000). Yield potential on a per plant basis and determined under optimal resource capture conditions, has not changed (Duvick, 1997; Tollenaar and Lee, 2002). The proportion of total dry matter allocated to the grain (i.e., HI) has not changed (Duvick, 2005). Plant growth rate at silking has not changed much (Crosbie, 1982) and DMA up to silking has not changed (Tollenaar et al., 1994a).
Results of the ERA-hybrid studies showed a 2.13-fold increase in grain yield between 1930s and 1990s hybrids (Duvick et al., 2004). The apparent discrepancy between the sixfold increase in on-farm U.S. maize yield and the 2.13-fold increase in the ERA-hybrid studies is attributable, in part, to the absolute yield levels: 1.3 and 7.2 Mg ha–1 for on-farm yield in 1939 and 2000, respectively, and 4.8 and 10.2 Mg ha–1 for 1930s and 2000s ERA-hybrids, respectively (Duvick et al., 2004). Since HI did not change during this period, the 2.13-fold increase in ERA-hybrid grain yield represents a 113% improvement in DMA. The increase in DMA (i.e., the "source") can be attributed, in part, to quantifiable changes in light interception due to increased leaf area index (LAI) and changes in light utilization due to more erect upper leaves (Fig. 3a and 3b). Another part of the improvement in DMA is attributable to maintenance of green leaf area (Fig. 3c and 3d) and leaf photosynthesis during the GFP and this portion of the contribution to improvement in DMA cannot be directly quantified. The contribution can be estimated, however, from the difference between the total increase in DMA and the contribution of increased light interception and canopy architecture to increased DMA.
Quantifiable Effects of Light Interception and Utilization.
Seasonal DMA is a function of the duration of the life cycle, and the interception and utilization of incident solar radiation throughout the life cycle. Light interception is primarily driven by leaf area, while light utilization is a function of canopy photosynthesis. Leaf area per plant has remained fairly stable during the hybrid era (Crosbie, 1982; Duvick, 1997), but increased plant population density tolerance has effectively increased leaf area index (LAI) from
2.4 m2 m–2 for a 1930s hybrid to
4.8 m2 m–2 for a 2000s hybrid. Doubling the LAI from 1930s to 2000s hybrids has translated into an approximately 20% greater light interception (Table 1
). The effect of greater LAI on light interception is offset, in part, by the steeper leaf angle of newer vs. older hybrids (Duvick et al., 2004; Fig. 3a and 3b). The combination of greater LAI and steeper leaf angle has resulted in an approximately 14% increase in light interception of newer vs. older hybrids (Table 1). The utilization efficiency of intercepted irradiance at high levels of photosynthetic photon flux density (PPFD) increases when leaf angle of the canopy becomes steeper because of the more even distribution of light within the crop canopy and the curvilinear nature of the photosynthesis–light response curve (e.g., Long et al., 2006). Tollenaar and Dwyer (1999) estimated that the upper limit for the increase in canopy photosynthesis due to an increase in leaf angle from 30 to 60° ranged from 15 to 30%, which translates to an approximately 15% increase in seasonal DMA. The benefit of increased leaf angle on canopy photosynthesis can be realized only when light interception is high (i.e., at a high LAI), because a reduction in light interception due to increased leaf angle will frequently outweigh the benefit of a more even light distribution at a relatively low LAI. Consequently, the association between the more erect leaf angle and yield improvement in maize is probably a result of, and not a consequence of, increased plant density tolerance. The combined contribution of increased LAI and increased leaf angle on the increased DMA is 31% (i.e., 1.14 x 1.15 = 1.31).
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Sink size in maize is a function of kernel number and kernel weight. The genetic improvement in grain yield is highly associated with increased kernel number, rather than kernel size or kernel weight (Tollenaar et al., 1992; D. Duvick, personal communication, 2005). Kernel number in maize is a function of rate of DMA during a period bracketing silking, in other words plant growth rate (g DM d–1 plant–1) (Edmeades and Daynard, 1979; Tollenaar et al., 1992; Fig. 6 ). Rate of DMA during the period bracketing silking varies little among older and newer maize hybrids (Crosbie, 1982; Tollenaar et al., 1994a). Consequently it can be deduced that the increase in kernel number in newer hybrids is a result of greater partitioning of dry matter to the kernels during the sensitive period of kernel establishment. Support for this contention is presented in results with Argentine maize hybrids (Echarte et al., 2004) and comparison among hybrids and their inbred lines (Echarte and Tollenaar, 2006).
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50% even when grown under stress conditions. It is important to realize that improvements in source components could only result in increased grain yield if there were simultaneous improvements occurring in the sink components (i.e., increasing DMA without increasing grain sink size will not lead to increased yield). The balance between source and sink is maintained in newer hybrids grown across a range of plant population from 30,000 to 79,000 plants ha–1, whereas in older hybrids the sink/source ratio declines with increasing plant populations (i.e., the decline in HI).
Overview of Modern Maize Breeding
Hybrid maize traces its roots back to experiments on heterosis and inbreeding conducted by G.H. Shull (1908, 1909) at Cold Spring Harbor Laboratories in New York and E.M. East (1909) at Connecticut State College. Those observations made nearly 100 yr ago, and methodology outlined by Shull (1909) gave rise to the modern hybrid maize industry (cf., Crow, 1998). Because of the hybrid nature of the crop modern temperate maize breeding in the United States and Canada has evolved into two very distinct activities: inbred line development and hybrid commercialization (Duvick and Cassman, 1999; Fig. 7
). It is important to understand the distinctions between the two breeding activities, as genetic improvements first must occur within the inbred line development programs before they can be captured and realized in commercial hybrids. Inbred line development is the stage of maize breeding where the greatest amount of de novo genetic variation is present, created through recombination giving rise to novel alleles and new allelic combinations. In hybrid commercialization the genetic variation is potentially less, but represented by a far more refined germplasm pool; one that has been through extensive evaluation.
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Inbred Line Development
The majority of inbred development activities in North America involve the use of the pedigree method of breeding (Duvick et al., 2004; Mikel and Dudley, 2006) (Fig. 9
). Pedigree breeding as it is structured in the commercial sector is akin to reciprocal recurrent selection (RRS). Breeding crosses tend to be made by crossing inbred lines within a heterotic pattern. Inbred lines from the other heterotic patterns are used to improve the heterotic pattern represented by the breeding cross. This type of breeding scheme (i.e., akin to RRS) allows maize breeders to improve both additive and nonadditive genetic effects, resulting in greater overall genetic gains (Comstock et al., 1949). The typical pedigree breeding scheme generally consists of a two-parent breeding cross within a heterotic pattern. Parent selection is based on proven commercial utility of the inbred lines. An F2 population is formed from the breeding cross. Inbreeding is performed for several generations (e.g., S2) using ear-to-row with each family tracing back to different F2 plants. During the inbreeding process, genotypes with obvious defects are eliminated. Early generation testing occurs around the S2 generation, which involves forming topcross hybrids between the S2 lines and an inbred line from each of the main heterotic patterns. The resulting topcross hybrids are evaluated in a limited number of environments and selections based on agronomic performance are made. Only S2 lines that correspond to the selected topcross hybrids will be retained in the breeding program. The selected S2 families are further inbred to the S5 generation where a second round of topcross hybrid evaluation is performed. Again an inbred line from each of the main heterotic patterns is used to form the topcross hybrids. The hybrids are evaluated in a limited number of environments and selections are based on agronomic performance compared to commercial hybrids (i.e., checks). In general, all testing during inbred line development is done in hybrid combinations, involving relatively limited number of hybrid combinations, in relatively limited number of environments, and focused primarily on grain yield. At this stage any superior inbred lines are then considered for release to the hybrid commercialization side of the breeding activities.
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Future Prospects for Genetic Improvement
How will breeders achieve future gains in grain yield using conventional breeding approaches within commercial temperate maize germplasm? In an attempt to answer this question, we will examine the source and sink components (Fig. 2) disregarding their contribution to previous genetic improvement, by assessing their potential in terms of biological limits to selection and presence of genetic variation in the commercial germplasm pool. On the source side: flowering date and length of the GFP, leaf area and leaf angle, potential photosynthesis, and visual and functional stay-green will be examined; while on the sink side the dynamics between plant growth rate at silking and kernel number will be considered.
Flowering Date and Number of Days during the Grain-Filling Period
The GFP in maize is approximately 8 wk in length. As previously mentioned, all of the dry matter that is allocated to the grain is fixed by the maize plant during this period (Below et al., 1981; Cliquet et al., 1990). This is different than wheat (Triticum aestivum L.), for example, where dry matter fixed before anthesis can be remobilized to the grain (Rawson and Evans, 1971; Hans, 1993; Gebbing and Schnyder, 1999). If the GFP was lengthened to 9 or 10 wk, then theoretically more dry matter would be available for allocation to the grain. This would require anthesis and silking to occur earlier in the life cycle of the maize plant. Flowering date differences are quite substantial in the maize germplasm pool (e.g., 45–135 d; Gouesnard et al., 2002), however, the differences tend to influence the length of the life cycle of the maize plant rather than shifting when flowering occurs within the life cycle (Irish and Nelson, 1991; Corke and Kannenberg, 1989). Additionally, earlier flowering maize plants are smaller and have fewer leaves, resulting in a lower LAI. Therefore, the LAI–flowering date paradigm would need to be broken. Finally, greater sink capacity to accommodate the additional 1 to 2 wk of DMA and better functional stay-green (i.e., photosynthesis during the GFP) will be required to exploit this opportunity. While there is no evidence for biological limits to exist for any of these parameters, several paradigms need to be broken and all of these parameters need to be simultaneously altered to realize a favorable outcome (i.e., both source and sink size have to increase to increase grain yield). Given the complexity of this undertaking, it is probably not a reasonable target for conventional breeding approaches and therefore does not represent a great opportunity for future improvement.
Leaf Area and Leaf Angle
Each of these component processes accounted for
15% of the total improvement in DMA during the hybrid era (see Table 1). Currently, a typical maize crop intercepts
95% of the incident solar irradiation at silking with the existing canopy architecture and plant population densities (Tollenaar and Lee, 2006). The benefit of more erect canopy architecture is a better light distribution leading to higher canopy photosynthetic rates. The potential of future genetic gains in grain yield through increasing leaf angle any further is limited, at best (Fig. 3b). Therefore, further enhancements of leaf area or leaf angle does not present a reasonable opportunity for future improvement.
Visual Stay-Green
Current commercial genotypes are capable of retaining green leaf area beyond physiological maturity (i.e., black layer formation) (Fig. 3d). Once black layer is formed there is no biological reason for maintaining green leaf area. In point of fact, maintaining green leaf area beyond maturity may actually hamper moisture loss from the grain (Cavalieri and Smith, 1985). While it is extremely important that the current level of visual stay-green is maintained, there do not appear to be opportunities for further enhancement.
Potential Leaf Photosynthesis
The maximum level of leaf photosynthesis on a per unit area basis in maize occurs between full leaf expansion and silking. Potential photosynthesis levels have not changed during the hybrid era (Tollenaar et al., 2000) nor are they influenced by heterosis (Ahmadzadeh et al., 2004). Maximum net leaf photosynthetic rates at near saturating light conditions (i.e., PPFD = 2000 µmol m–2 s–1) in our studies with maize during the silking period have been in the range 45 to 55 µmol CO2 m–2 s–1 (i.e., approximately 40 mol photons mol–1 CO2 fixed). What is the theoretical upper limit for CO2 fixation in maize? The maximum quantum requirement (i.e., mol photons mol–1 CO2 fixed) at low light for maize has been estimated by Ehleringer and Pearcy (1983) as 15.0, which was very close to their observed maximum value of 16.1 mol photons mol–1 CO2 fixed. Consequently, the actual efficiency is 0.93 times the potential efficiency at low light levels. For potential leaf photosynthetic rate at near-saturating levels of incident PPFD, the estimate should include corrections for PPFD absorption by the leaf, respiration, and other sinks in the leaf that function as electron acceptors. The proportion of PPFD absorbed by the leaf is 0.9 (Earl and Tollenaar, 1997). Respiration in a fully grown maize leaf consists of maintenance respiration, which has been estimated as 1.5 to 2.0 µmol CO2 m–2 s–1 at 25°C by Earl and Tollenaar (1998), and includes also the energy associated with the conversion of the initial product of photosynthesis, glucose, to the substrate that is exported out of the leaf, sucrose (i.e., 0.92 g sucrose g–1 glucose). In addition, nitrate reduction in the leaf may serve as a sink for electrons generated by light reactions of photosynthesis, but those costs maybe negligible at saturating levels of PPFD when the dark reactions limit rate of photosynthesis. Hence, at a leaf photosynthetic rate of 50 µmol CO2 m–2 s–1 at 2000 µmol m–2 s–1 PPFD, the leaf is operating at approximately 50% of its potential in terms of the conversion of light energy into sucrose (i.e., 16.1/40 x 1/0.9 x 50/48 x 1/0.92). Leaves in a crop canopy will receive levels of PPFD less than 2000 µmol m–2 s–1 for a large portion of the day, because the sun–leaf angle will most of the time be <90° and partial shading by other leaves in the canopy will reduce the incident PPFD per unit leaf area, consequently, the mean daily efficiency of a leaf will be substantial >50% of its potential. Hence, it is not clear, whether the ratio between actual and theoretical maximum photosynthetic efficiency (i.e., 0.93) and/or the ratio of quantum requirement at high and low PPFD (i.e., 0.5) represent opportunities for improvement in maize leaf photosynthesis. Also, there is no evidence of favorable genetic variation in the germplasm pool (Ahmadzadeh et al., 2004). Therefore, even if increased leaf photosynthetic rate represents a opportunity for further improvement in source, the lack of favorable genetic variation within the germplasm pool means that it does not present an opportunity for conventional breeding approaches. In a review paper on selectable traits to increase yield of grain crops, Richards (2000) also suggested that altering rate of potential leaf photosynthesis does not offer a great opportunity.
Functional Stay-Green
Although the leaves in modern maize genotypes stay-green through physiological maturity, leaf photosynthetic rates decline during the GFP. While the decline in leaf photosynthetic rates is smaller in newer rather than older hybrids, the decline in leaf photosynthesis from silking through 6 weeks after silking is still about 50% (Fig. 5; Ying et al., 2000). This 50% decline in photosynthesis presents an opportunity that would translate directly into more dry matter being accumulated. The potential for further genetic progress through plant breeding is great, as favorable genetic variation is still present in the germplasm pool (Ahmadzadeh et al., 2004).
Sink Establishment
Any increase in sink size will be the result of increasing kernel number. Kernel number can be changed by either increasing the rate of DMA or by a change in partitioning of dry matter during the sensitive period of kernel establishment (i.e., the period bracketing silking). Since light interception is maximized during silking and the maximum rate of leaf photosynthesis is not likely to increase, the probability of increasing the rate of DMA during silking is negligible. Consequently, any further enhancement in kernel number should be the result of increased dry matter partitioning to the kernels during the sensitive period of kernel establishment. Evidence of genetic variation for sink establishment parameters is evident in both the North American germplasm pool (e.g., Echarte and Tollenaar, 2006; Fig. 6) and the Argentine germplasm pool (D'Andrea et al., 2006), suggesting that opportunities do exist for further genetic progress in enhancement of sink establishment.
How to Capture these Opportunities— Build on What Has Worked in the Past
The use of high plant population densities as a selection tool allows the breeder to simultaneously apply stress to source processes as well as sink processes. The plant literature is rich with many references to stress and the area of abiotic stress tolerance has been the focus of a tremendous amount of research, particularly at the molecular level (e.g., Cushman and Bohnert, 2000; Seki et al., 2003). Stress can be defined as either acute or chronic exposure of the plant to unfavorable environmental conditions; however, stresses do not have to be extreme to elicit responses from the plant or to have an impact on productivity of the plant, nor do they need to be attributed to one single unfavorable environmental component. Stress in the context of experimental plant biology is usually viewed as being acute (i.e., extreme) deviations in temperature, salinity, or water status from the ranges normally experienced by the plant. When working with these types of specific, acute stresses the range of stress levels that result in discernible variation is quite narrow. Plant population density stress, on the other hand, is a chronic stress that results from potentially several simultaneous unfavorable environmental attributes which reduces either resource capture or utilization (Tollenaar and Lee, 2002). Plant density is a moderate stress that is easy to manage in its intensity; it is suitable for high-throughput scenarios; it affects the plant throughout its life cycle; and the range of stress levels that result in discernible variation is quite large. The individual environmental components that make up "plant population density stress" are hard to define, but as an aggregate they can be defined as chronic deviations from optimum growing conditions that limit the plant from attaining its genetic potential (Fig. 3h and 3i). This is similar to the definition used by Boyer (1982) which defined stress environments as those that prevent the plant from reaching its full genetic potential for reproduction. In this context it has been estimated that, stress (i.e., less than favorable environments) reduces the reproductive genetic potential of crop plants in the United States by an average of 70% {e.g., 66% for maize, 82% for wheat, 69% for soybean [Glycine max (L.) Merr.], and 81% for sorghum [Sorghum spp.]) (Boyer, 1982). Interestingly, the "gap" that remains to be filled between realized yield and yield potential in modern maize hybrids is also 66% (Tollenaar and Lee, 2002).
What stage should plant density be incorporated into a corn breeding program? Current commercial production maize plant population densities in North America are in the range of 75,000 to 80,000 plants ha–1 under favorable agronomic conditions. Most modern commercial maize breeding programs are using high plant population densities (>160,000 plants ha–1) to evaluate germplasm. But this evaluation is being done during the advanced phases of hybrid commercialization. To drive the genetic gain equation, directly selecting for a trait in a segregating population is always superior to only choosing parents that possess the favorable phenotype. As it is structured now, high plant population density testing during hybrid commercialization at best will enable density tolerant parents to be chosen for the breeding crosses. We propose that high plant population density trials need to be incorporated into inbred line development. The best use of resources would be to introduce this into the advanced testing phase of inbred lines. And we also advocate coupling performance of the S5 lines in testcrosses at high densities with performance at conventional densities when making the final selection decisions (Fig. 9). By incorporating this into the inbred line development programs and across all of the main heterotic patterns, both additive and nonadditive genetic effects influencing stress tolerance can be improved on.
The genetic improvement over the past 65 to 70 yr is the result of changing physiological attributes of the corn plant. Those physiological attributes have imparted enhanced abiotic stress tolerance. However, throughout the hybrid era the only trait that primarily drove selection was grain yield, not focusing on a particular physiological process, mechanism, or stress response. Measuring grain yield, from a physiology perspective is the ultimate selection index for optimizing the physiology of grain yield. The sixfold increase in commercial grain yield during this period of time is the result of focusing on grain yield, in a long-term breeding scheme that was constantly challenging the physiology of the corn plant to maximize grain yield. Future gains will no doubt come about through continued evolution of breeding methodologies and philosophies. The changes need to occur within the inbred line development side of the program and they most likely need to occur within all of the heterotic groups.
Received for publication April 10, 2007.
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