In silico design of crop ideotypes under a wide range of water availability

Abstract

Given the changing climate and increasing impact of agriculture on global resources, it is important to identify phenotypes which are global and sustainable optima. Here, an in silico framework is constructed by coupling evolutionary optimization with thermodynamically sound crop physiology, and its ability to rationally design phenotypes with maximum productivity is demonstrated, within well‐defined limits on water availability.  Results reveal that in mesic environments, such as the North American Midwest, and semi‐arid environments, such as Colorado, phenotypes optimized for maximum productivity and survival under drought are similar to those with maximum productivity under irrigated conditions. In hot and dry environments like California, phenotypes adapted to drought produce 40% lower yields when irrigated compared to those optimized for irrigation. In all three representative environments, the trade‐off between productivity under drought versus that under irrigation was shallow, justifying a successful strategy of breeding crops combining best productivity under irrigation and close to best productivity under drought.

INTRODUCTION

The primary limitation of agriculture in both developed and developing countries is the large variability (or lack) of rainfall (Wilhite & Glantz, ). Despite numerous breeding efforts (Zamir, ), productive agriculture still requires large quantities of water that are then simply lost to the atmosphere through plant stomata. There are clear, achievable strategies to improving agricultural water use—reducing pre‐irrigation water losses, optimizing irrigation delivery (Geerts & Raes, ), and breeding crops with reproductive sensitivities to drought (Bartels & Sunkar, ; Ingram & Bartels, ; Kasuga, Liu, Miura, Yamaguchi‐Shinozaki, & Shinozaki, )—but changing the actual water use of the plant has met limited success to date (Geerts & Raes, ). Traditionally, genetic hybridization of superior crop varieties is performed to breed more productive maize without explicit direction from physiological hypotheses. As water stress has a large effect on reproductive productivity (Flexas & Medrano, ; Tezara, Mitchell, Driscoll, & Lawlor, ), one option for breeding crops is to alter the use of available water thereby avoiding water stress and maintaining productivity over long periods between rainfall. Here, we ask whether conservative water use by crops is feasible and at what expense in terms of yield under both irrigated and drought conditions.

Improved performance of crops under water stress (Araus, Slafer, Reynolds, & Royo, ), and conservative or efficient water use would have a huge impact on the sustainability of agriculture, especially in the 1.3 billion hectares of marginal agricultural land (Post et al., ) available globally. In arid regions where irrigation water is expensive, conservative crop water use would allow the cultivation of a wider range of crops, even with minor decreases in water use. Furthermore, conservative water use would be beneficial as it would increase the probability of crops surviving dry periods that typically occur during the growing season (Sinclair, Marrou, Soltani, Vadez, & Chandolu, ). As an emblematic species of crop productivity, we ask in this (in silico) study how maize can be bred to achieve altered water use, in three representative environments, and what ideal characteristics (the “ideotype”) do breeders have to select for to do so.

The hypothesis‐driven approach of “ideotype breeding” is the process of first defining what an ideal plant should look like (the “ideotype”) and then creating such a plant. “Ideotype breeding” per se is not new. It is typically focused on the assumption that a few traits are key to improving plants, based either on intuition (Sedgley, ) or on experimental data (Lynch, ). Similarly, many physiological models of crops (Dauzat, Rapidel, & Berger, ; Doussan, Pagès, & Vercambre, ; Duursma & Medlyn, ; Javaux, Couvreur, Vanderborght, & Vereecken, ; Lai & Katul, ; Lhomme, Rocheteau, Ourcival, & Rambal, ; Manzoni, Vico, Palmroth, Porporato, & Katul, ; Rings et al., ; Somma, Hopmans, & Clausnitzer, ; Sperry, Hacke, Oren, & Comstock, ; Thompson, Zwieniecki, & Holbrook, ; Thornley, ; Vogel, Dohnal, Dusek, Votrubova, & Tesar, ; Woo, Boersma, & Stone, ) have been developed since the pioneering 1948 water relations model of van den Honert (Honert, ), focusing on specific aspects of plant physiology—enumerated hereafter in order of frequency: water transport, time dependence, influence of environmental conditions, heat and mass transfer, effect of plant geometry, nutrient transport, plant growth, and phloem transport. While these models can determine and explain optimum relationships between existing traits (Jensen et al., ; Manzoni et al., ; Rings et al., ), it still remains an open challenge to leverage this knowledge to direct breeding efforts. Recently limited works (Alarcon & Sassenrath, ; Ding et al., ; Drewry, Kumar, & Long, ; Wang et al., ) have been done in integrating plant physiological model to numerical optimization to direct breeding efforts. The focus of our work is to integrate a plant physiology computational model with an evolutionary algorithm that identifies optimal crop (hydraulic) characteristics. This framework allows the evaluation of crops and their optimization (a computational equivalent of breeding) in silico, using high‐performance computing resources. The resulting ideotypes are identified with minimal input, assumptions, or intuitions from the “numerical breeder,” and are compared to the ideotypes obtained from traditional breeding strategies. This in silico integration of an optimization algorithm with a plant physiology model might offer value addition to the traditional breeding strategies mentioned above.

MATERIALS AND METHODS

Plant physiology model and implementation

This model is based upon a detailed 1‐dimensional representation of plant hydraulic characteristics, as represented in Figure . Liquid‐phase plant‐water relations are represented as a static series of conductances/resistances for leaves, stems, and roots without capacitance, as in the seminal work of van den Honert (). The model is a physiologically explicit representation of C4 maize water use after canopy closure. The model explicitly accounts for energy balance (convection, radiation, and latent heat), transpiration, intercellular CO₂ concentration (via both diffusion and biochemical processes), weather conditions (temperature, precipitation, pressure, and radiation), and soil type. The energy balance, transpiration, and intercellular CO₂ concentration are implemented via different submodels. Figure shows the inputs, outputs, and submodels of the model. A full description of the model, as well as a detailed description of its calibration, verification of thermodynamic soundness, and validation is available in the Appendix 1. The model can predict photosynthetic rates, leaf transpiration rates, and survival probability as a function of different combinations of phenotypic traits.

A conventional resistance or conductance (resistance = 1/conductance) model of maize hydraulics (panel a) and the model used for simulating maize hydraulics including feedbacks (panel b). (panel a) Loss of the water to the environment (Transpiration rate (Tr)) is proportional to the difference in water potential between soil and air, and inversely proportional to conductances in series: Ksoil, Kroot, Kshoot (Kstem + leaf), boundary layer conductance, and conductance of stomata. (panel b) The conductance of the stomata to water vapor (gst) and CO2 is modulated by the water potential of the leaf (ψleaf) when it is below a threshold (ψth). The conductance (gst) modulates between maximum stomatal conductance (gmax; a proxy of how many stomata and how wide they open), which sets the maximum water loss rate (transpiration rate) and the maximum CO2 uptake rate (~photosynthetic rate) for sunlit leaves, and minimum stomatal conductance, gmin, which affects the rate of desiccation under drought, but this state also prevents CO2 uptake. The effectiveness of the modulation (slope of the response curve) is tuned by an inherent sensitivity (Sl) or a contribution of the root, based upon the sensing of soil drying (Sr)

Model inputs, outputs, and connectivity among the submodels. Description of the symbols can be found in the Appendix 1

Plant photosynthetic rate is dependent upon stomatal pores in the leaf being open to allow CO₂ to diffuse into the leaf, but this is unavoidably at the expense of water loss out of the leaf. That is, hundreds more H₂O molecules are lost to the air than the few CO₂ molecules that enter. Thus, it seems likely that breeding strategies that maximize productivity, or CO₂ uptake, would occur by having stomata open more of the time on average, at the expense of increasing water use. However, plant water flow is a rich process, with many traits that may be “tuned” to enable the plant to optimize water use relative to CO₂ uptake. Seven plant hydraulic “traits” are considered in the model of plant hydraulics, shown in Figure , and can be used to represent the response of leaf transpiration to environmental variation.

The physiological model is implemented in MATLAB with inputs of soil and hourly weather data over a 61‐day (~1,460 hr) period. Each model evaluation for a given trait configuration—producing hourly outputs—took about 40 s on a standard laptop.

Three hypotheses for ideotype design

By keeping the other phenotypic traits of the maize plant constant, the hydraulic traits can be varied to design in silico plants adapted to irrigated conditions and/or to the limited water volumes available under drought. Note that these traits represent the adult crop and do not represent the reproductive responses of maize to drought. To study the latter would require a further set of breeding characteristics that describe maize adaptation in dry environments where water stress does occur.

To quantify the trade‐off of crop productivity under drought versus irrigated conditions, we define two quantities, the drought productivity gap (the difference of total photosynthetic yield between a plant optimized for yield under irrigated conditions versus a plant optimized for yield under drought, when both are grown under drought) and the irrigated productivity gap (the difference of yield between a plant optimized for yield under drought versus a plant optimized for yield under irrigated conditions, when both are grown under irrigated conditions). See also the discussion of Figure ; Table . A number of hypotheses were expressed using these concepts of trade‐offs:

Cumulative photosynthetic productivity (Photo = Δt∑i=1NPR) (with simulation uncertainty ± 12 mol/m2) of two million simulated maize “trait” configurations (panel a), created by varying seven plant hydraulic traits in an irrigated environment and the same environment under drought, that is, without irrigation (Davis, CA, USA). The trait variation for four traits is shown in panels b–e for four of the groups of configurations on the edge of the trait space, as defined in panel a. Unfilled symbols represent configurations that survived two months of drought. The configurations were created through random combinations of traits or were the result of a genetic algorithm to find the highest productivities under drought or irrigation. Ideotypes were defined as the trait configuration at the intersection of genotype groups 2–3 (drought), 3–4 (smart), and the highest productivity trait combination 4–5 (extravagant). Representative configurations of drought (triangle), smart (circle), and extravagant (star) along with stingy (between groups 1 and 2, square) are referred to below. Mean and standard deviation of the traits of some selected configurations adjacent to the representative configurations (ideotypes) are presented in Table

Mean and standard deviation of the hydraulic traits of the selected configurations near the representative configurations (ideotypes) as shown in Figure . For these configurations, the cumulative photosynthetic productivities under irrigated and drought conditions are within 10% of those for the ideotypes

Adjacent ideotype	g min	g max	Sl(θl)	ψ th	Sr(θr)	K shoot	K root
Stingy	0.024 ± 0.01	0.34 ± 0.50	42.82 ± 30.2	−0.13 ± 0.07	54.84 ± 31.46	6.82 ± 2.58	22.18 ± 28.18
Drought	0.017 ± 0.004	0.36 ± 0.33	50.99 ± 12.32	−0.32 ± 0.11	50.19 ± 16.67	7.69 ± 2.54	9.56 ± 6
Smart	0.016 ± 0	0.35 ± 0.15	35.75 ± 17.3	−0.22 ± 0.04	71.7 ± 19.47	19.73 ± 11.17	60.03 ± 11.8
Extravagant	0.135 ± 0.025	1.34 ± 0.59	26.26 ± 19.16	−0.68 ± 0.28	32.62 ± 18.6	37.71 ± 7.87	52.13 ± 13.04

Hypothesis 1: A strong trade‐off, that is, a large irrigated productivity gap, exists where drought‐adapted plants have traits leading to water conservation and thus have severely limited maximum productivity under irrigated conditions.

Hypothesis 2: Smart plants can be bred. Plants with traits that lead to high sensitivity to water deficits might be able to offer some intermediate strategy where the plant is moderately adapted for both drought and irrigated conditions—a smart plant. Smart plants exhibit small drought productivity gaps, as well as small irrigated productivity gaps. Effectively, these results will indicate whether a single trait combination can robustly perform in both irrigated and drought‐prone environments.

Hypothesis 3: The impact of increasing atmospheric evaporative demand, temperature, and solar radiation between sites along a mesic to arid transect of US maize‐growing regions (Iowa < Colorado <California, Supporting Information Figure S1) will lead to a stronger trade‐off between productivity and drought survival in more arid regions. That corresponds to both the drought and irrigated productivity gaps being larger in arid than mesic environments.

Framework to assess the hypothesis

Simulations were performed using weather and soil data for a representative two‐month period of June and July 2010 for three sites representative of the aridity gradient in the United States: Ames, Iowa; Greeley, Colorado; and Davis, California. The physiology model was integrated into a numerical optimization framework to determine which combination of seven plant hydraulic traits resulted in optimization of two extreme goals: (Goal 1) Maximum productivity (estimated as cumulative sum of photosynthesis) in an irrigated environment where evapotranspired water loss was matched by irrigation and (Goal 2) Drought survival and productivity during two months of drought—an entire lack of rainfall.

The optimization algorithm used was a parallel implementation of a genetic algorithm which is a metaheuristic evolutionary algorithm (Schneider & Kirkpatrick, ). The algorithm evaluates the “fitness” or cost function (defined below) of several hundred trait combinations (configurations) per “generation,” where evaluation of one trait configuration is computed by one processor. The algorithm identifies optimized traits that maximize the cost function by exploring the space of trait configurations across several hundreds of generations. The choice of using a metaheuristic (stochastic), gradient‐free optimization algorithm is motivated by two reasons: (a) the cost function is highly corrugated with many local minima (see Supporting Information Figure S2), thus precluding the use of any gradient‐based, single‐start algorithm, and (b) we are interested in not necessarily individual minima, but in identifying ideotypes (that are defined by trends that the solutions exhibit). Thus, a stochastic evolutionary algorithm with multistart capabilities provided several reasonable trait configurations that could be analyzed to infer trends, identify any Pareto front (Manzoni et al., ), and group identified trait combinations into ideotype classes.

A parallel implementation of the genetic algorithm (GA) was used to search for trait combinations that maximized the cost functions for Goal 1 and Goal 2. The cost function used was a weighted combination of productivity under irrigation and productivity under drought. The MATLAB® implementation of the model is integrated with the GA framework available in MATLAB® and deployed on the computing clusters available at Iowa State (CyEnce cluster) and via NSF XSEDE resources (TACC Stampede). Supporting Information Figure S3 shows the flowchart of the model integration with the GA framework. The simulations usually took about 4 hrs to run for each optimization run on a server with 16‐core 2.0 GHz processor with 128 GB RAM. Optimizations were initialized with different random seeds and rerun ten times to get statistically consistent results. The Pareto front was explored by using a sequence of weights for the two goals. This is defined in the cost function as follows:[Image Omitted. See PDF]where $\text{PR}=f\left(g_{min},g_{max},S_l,S_r,{\psi }_{\mathit{th}},K_{\text{shoot}},K_{\text{root}}\right)$ is the output (photosynthetic yield rate) from the model for a specific trait configuration. Here, ∆t and N are time resolution in the weather data and a total number of hours, respectively. θ defines the relative weight for the first and second terms. The first term tracks response under irrigated conditions, while the second term tracks response under drought conditions. Six values of θ equally spaced from 0 to 90° were used to construct the Pareto front. The drought and irrigated productivity gaps offer an insightful way to reason about the Pareto front. Each optimization was repeated ten times to ensure proper exploration of the trait space. Over three million distinct trait combinations were evaluated across three geographic sites.

The source code for the framework including the plant physiological model, the optimization routine, and postprocessing modules is available at https://bitbucket.org/baskargroup/cropdesign/src/master/.

RESULTS AND DISCUSSION

Trade‐off between adaption to drought and irrigation

In the aridest environment (California; Figure ), different configurations of maize hydraulic traits led to a wide range of performance under drought or irrigated conditions. Most trait configurations were either equally or more productive under irrigation than drought, with an envelope bounding the possible configurations (colored points; Figure ). Configurations that had maximum productivity under drought (intersection of groups 2 and 3 in Figure ) corresponded with an approximate halving in productivity under irrigation relative to trait combinations with maximal productivity. Moderate‐to‐high irrigated productivity trait combinations (intersection of groups 3 and 4) retained high productivity under drought, and some configurations near this intersection survived drought (unfilled points). Thus, at the extremes of selection for maximum productivity (group 5) or drought survival (intersection of groups 2 and 3), there is a strong trade‐off in productivity between the two environmental conditions. This verifies hypothesis (1) and corresponds to a large irrigated productivity gap. But for intermediate trait configurations (group 3), a middle ground exists that allows selection for moderate–high productivity in both environments. The existence of an area of possible trait combinations above the straight line connecting the ideotypes for drought and for irrigated conditions tends to verify hypothesis (2) that a single ideotype can be applied to irrigated and drought‐prone environments.

Productive trait configurations

Configurations leading to extreme productivity under irrigation (groups 4 and 5) had a large shift toward greater potential for stomatal opening (Figure b,c) and poor performance (do not survive, filled symbol) under drought. The configurations with the highest productivity under drought had thresholds for stomatal closure corresponding to relatively negative soil water potentials (Figure d; water potentials of soil or components of a plant indicate how much the plant has to “suck”—the tension—to pull water from the soil and through the plant). The total plant conductance of water increased with plants maximally productive under irrigation (Figure e). The sensitivity of the stomata to leaf water status was orthogonal to that of root water status, that is, the two traits can substitute for each other (Supporting Information Figure S4), but high sensitivity to both root and leaf water status was not found among configurations along the envelope (Deb, ). A few unnatural trait configurations were much more productive under drought than irrigation due to a combination of traits that led to death under irrigated conditions, but not under drought (group 6, see Appendix 1; Table for an extended discussion).

Hydraulic traits of a typical configuration in group 6

g min	g max	Sl(θl)	Sr(θr)	ψ th	K shoot	K root
0.04	0.31	12	88	−0.57	4.2	97

Ideotypes for performance under different water availabilities

Using the results of the California study (Figure ), we define three ideotypes based on productive trait combinations for environments with different water availabilities. A smart ideotype maintains open stomata during mesic periods, but rapidly responds to water deficit, conserving water from that point onwards. The traits that led to this were moderate values for maximum stomatal conductance and switching to low minimum conductance value under drought due to a combination of high sensitivity to stress and a high threshold (early closure during the dry down) for stomata closure (Figure ). In contrast, the extravagant ideotype had a maximal stomatal opening at all times with little sensitivity of stomata to close under stress (Supporting Information Figure S5). This allowed the plants to maximize productivity at the expense of maximal water use but led to poor survival under water‐limited conditions. The drought ideotype had higher plant hydraulic resistances, leading to greater sensitivity to water status. Coordination of an intermediate threshold for stomatal closure with high sensitivities of the stomata to water stress led to successful closure under limiting conditions. Along with these ideotypes, we define a stingy type plant (which is not an ideotype). A stingy plant had traits that took water conservation to an extreme, effectively limiting productivity under all conditions. This type was unable to use all the water available to it due to limited maximal stomatal conductances and a threshold for stomatal closure that resulted in some closure under all conditions (Figure ).

Performances of four trait combinations under drought (Davis), showing typical configurations defined as ideotypes in Table and Figure (sky blue lines in (a–c) indicate atmospheric water potential). The extravagant plant, adapted for high yield under irrigation, started with extreme transpiration rates and ended with the wilting due to inability to control loss of water. The highest yielding plant under drought (drought ideotype) was able to keep stomata open to photosynthesize through the drought period due to initial water conservation, while the smart plant modulated this behavior from high to low water use over the soil dry down. The stingy plant was the most consistent throughout the dry down, but its initial conservative water use resulted in unused water by the end of the drought period and greatly limited its productivity under irrigated conditions

General rules for productive traits combinations

The influence of traits on water use can be explained as follows. The maximum stomatal conductance (g_max) sets the maximum CO₂ uptake rate of the plant and thus the productivity. Thus, irrigated agriculture would require plants with maximal g_max. In areas where water supply is limited, the marginal returns on opening stomata are small at high g_max, while an increase in water loss is high. For this reason, drought‐prone areas would require moderate g_max values effectively increasing water use efficiency through water conservation. As g_max is achieved under most well‐watered daytime conditions, this trait has a direct effect on plant performance that is unlikely to be highly interactive with other traits and thus deserves attention from breeders. In particular, stomatal density—a predictor of g_max—is easy to phenotype (Gitz & Baker, ).

The minimum stomatal conductance (g_min) sets the rate of water loss under severe soil water deficits, and thus, is important in drought‐prone environments (Sinclair, ) especially to conserve water toward the end of the growth cycle. However, daytime values of stomatal conductance reaching g_min would likely be a good indicator of reproductive failure due to desiccation, and thus, this trait enhances survival under drought, not necessarily yield. In irrigated environments, g_min is reached at night. Although the adaptive function of high stomatal conductance at night is unclear (Caird, Richards, & Donovan, ), many crops have this strategy (sorghum (Muchow & Sinclair, ), cotton (Fish & Earl, ), and soybean (Hufstetler, Boerma, Carter, & Earl, ). One result of the simulations was that plants selected for maximum productivity under irrigated conditions had extremely high g_min, and this was largely due to the decreased leaf temperature at night due to evaporation and thus reduced respiration rates. Cumulatively, this led to slightly higher productivities.

The benefit of the other traits (components of K_plant, S_r, S_l, and ψ_th) is largely interdependent, and thus, multiple trait combinations can result in similar behavior. A possible reason is that the traits are not always expressed always, for instance, stomatal closure according to the sensitivity (S_l) does not occur unless the water potential is below the ψ_th value, while the water potential value in turn is determined by the value for K_plant. General rules are evident though. Irrigated plant that had high productivity did not close stomata and thus the K_plant must be high proportional to the value of ψ_th so that leaf water potential was not in a range that would trigger stomatal closure. If such trait combinations are successful in preventing stomatal closure, then the sensitivities of the stomata play little role in determining performance in irrigated conditions.

Under drought conditions, however, the sensitivities of the stomata to the leaf or root water potentials are proportional to the difference between the value of ψ_th and permanent wilting point, for example, if the ψ_th is near the wilting point, then stomata have to be very sensitive to ensure full closure before wilting occurs. Again, K_plant needs to be proportional to the value of ψ_th, but generally on the lower end of the relationship so that K_plant is limiting to water transport leading to the leaf reaching the ψ_th upon which the stomata close and water is conserved. The stomatal sensitivities to root and leaf water status are mutually exclusive to each other, as drought‐adapted plants are sensitive to either root or leaf water status, not both (Supporting Information Figure S4). Both sensitivities result in similar stomatal closure behavior and water conservation but require different conductance within plant organs.

Performance at mesic sites

The shaded red and shaded green lines in Figure show the Pareto front of crops optimized for the respective environments of Iowa and Colorado. In both regions, the drought and irrigated productivity gaps are virtually zero, as indicated by the vertical slope of the line connecting the drought ideotype and the extravagant ideotype for those sites. This indicates that in both mesic environments, ideotypes designed for drought also perform best in irrigated conditions. This finding supports the hypothesis (3). Next, we investigate how ideotypes designed for the California environment (shown by geometric symbols in Figure ) perform in mesic environments. The extravagant ideotype, with the highest productivity under irrigation in California, retained the highest productivity in Colorado and Iowa as well (Figure a, stars). The shallow drought productivity gap in California (between the circle smart and triangle drought ideotypes; Figure ) was nonexistent (higher drought productivity gaps) in Iowa and Colorado (Figure ). Thus, the configuration with the highest productivity under irrigation in CA was transferrable to all environments and even performed reasonably well under drought in mesic environments. However, the configuration with maximal productivity under drought in California performed poorly at the other sites relative to the configuration bred for irrigated conditions. The higher evaporative demand in California (lower relative humidity, higher temperatures, and solar radiation) led to the smart trait configuration being able to survive a longer drought in the more mesic environments, with less stress and thus greater productivity.

Plant performance can be linked to the plant physiology, under irrigated conditions (top right) and under drought (bottom right). Plant performance is reported (left) for every configuration on the Pareto front (line) for California (Davis) and the representative configurations (symbol) nicknamed after their use of resources as stingy, drought, smart, or extravagant. The shaded thick lines show the performance of those configurations in Iowa (Ames; red) and Colorado (Greeley; green). The colors indicate the geographic site. To show that the extravagant plant dies under drought, a full rather than empty symbol is used. The red and green lines show the envelope of traits (a Pareto front) obtained for Iowa and Colorado, respectively. It is interesting to see that there is minimal variation between the results of explicit optimization under Iowa and Colorado conditions when compared with the results of “transplanting” the California environment ideotypes to Iowa and Colorado conditions. The panels on the right compare the physiology of the representative configurations under irrigated (top) and drought (bottom) conditions, with values of the water transport resistance, water potentials (average leaf water potential is shown as a full square dot and variations as a bar), and cumulative water losses. Values are for three typical weeks. Stomatal conductances are shown as maximum stomatal opening (gmax). The presented plant conductances (resistances = 1/Kplant) do not depend on location, while stomatal conductances are location‐dependent. Here, the resistances are presented in terms of Davis, CA, weather

Higher productivities than the smart ideotype had extreme trait configurations that are somewhat unreasonable (Figure ). The high productivity was at the dramatic expense of performance under drought due to excessive stomatal opening. This opening led to maximal water use, but marginal returns in the form of CO₂ uptake which saturates at high stomatal conductance. Thus, for very little return these plants used considerably more water and thus performed very poorly when water was limited. In this context, the smart ideotype performed well in both drought and irrigated environments and may be a satisfactory solution for environments where drought is not the normal situation. The performance of the smart ideotype, designed for California conditions, was excellent in the other two locations with productivities within 10% of the yield of ideotypes optimized for respective drought (highest points of the green and red shaded lines in the Figure a) or irrigated conditions (rightmost points of the green and red shaded lines in the Figure a) at these two specific locations.

One trait combination to rule them all (in most environments)

An unexpected finding is that, in the mesic Midwest (Iowa) and to some extent in the mountain environment (Colorado), the best performing ideotypes under irrigated conditions are the same as those under drought. In other words, there is practically no trade‐off in the US Corn Belt between breeding for yield under ideal or drought conditions. This promising finding echoes the conviction of Duvick, Smith, and Cooper (). Supporting Information Figure S6 in the supplementary documentation illustrates this finding, based on 80 years of breeding data. Based on Duvick's results, Cooper, Gho, Leafgren, Tang, and Messina () expressed his breeding philosophy as follows: the crop that performs best under favorable conditions is also the one which is most resistant to an array of adverse stresses, including water stresses. Similar results are evident in soybean (Specht et al., ). Note that both approaches, the one in this work and that of Duvick, are not entirely similar. Our work explores the phenotypic space of a single plant, with the purpose of maximizing its yield, while Duvick breeds hybrids with the purpose of maximizing the yield of an entire field. In that respect, Duvick showed that increases in yield per unit area of the newer hybrids are owed not to increased yield potential per plant but rather to the hybrids’ ability to grow at higher plant densities. The fascinating promise stemming from the conjunction of both approaches is that a combination of the breeder's ability to increase planting densities with our findings that single plant performance can be improved might accelerate the rate of yield gain in future breeding efforts.

Thus, breeding programs in mesic environments have less need to differentiate breeding efforts between efforts for irrigation and drought—one genotype could potentially do well in both. The selection for high productivity in mesic environments can be broad—applying to both drought and irrigated conditions, and also to arid irrigated environments. However, arid environments and environments that have sustained drought periods (i.e., longer than those simulated here) would require selection for trait configurations that were conservative of water use (the drought and stingy ideotypes). These genotypes would likely need to be adapted to local environments and would not be useful in irrigated environments (Figure ).

Flowchart for the implementation of plant physiology model

CONCLUSIONS

The above study describes how breeding efforts can be informed by a numerical framework integrating optimization of an ensemble of seven phenotypic traits and a comprehensive model for plant physiology. Over three million crop configurations were bred in silico for three environments to extract crops with traits that maximize productivity under a representative range of environments and irrigation conditions. The ideotypes identified are physiologically sound and do not require specific guidance nor input from the “numerical breeder.” Remarkably, we find that in mesic continental and semi‐arid environments, ideotypes optimized for drought also perform best in irrigated conditions. In hot and dry environments, the identified ideotypes are different for drought and irrigated conditions. However, in all three environments, the trade‐off between productivity under drought versus that under irrigation was found to be small. The existence of a minimum trade‐off corresponds to the existence of smart ideotypes, with identified traits such as high sensitivity to water stress and a high threshold for stomatal closure. Smart ideotypes have promising implications for maintaining our food supply under changing environmental conditions. The sheer amount and resolution of the data produced by the numerical framework (such as hourly variations of fluxes, temperature, and water potentials at various locations in the plant) open the way to real‐time comparisons with embedded sensors on crops (Figure ).

Temporal variation of stomatal opening, leaf water potential, and root water potential of a typical plant in group 6, both under well‐irrigated and drought conditions

The degree of flexibility that we have to adjust the relationship between crop productivity and water use determines sustainability in a changing climate and the global resource impact of agriculture. This study demonstrates the combinations of plant hydraulic traits that are needed to adapt maize to environments across the United States. In all environments, high yield was associated with maximum water use—a fact that is of great importance for global food supply. It is found that different environments require alternative breeding approaches, with some sites allowing pursuit of maximum productivity with limited consequence for drought performance.

ACKNOWLEDGMENTS

T.Z.J., B.G., and D.A. gratefully acknowledge financial support from the Presidential Initiative for Interdisciplinary Research of Iowa State University. B.G. and T.Z.J. gratefully acknowledge the Plant Science Institute at Iowa State University and computing support via NSF XSEDE CTS110007. M.E.G gratefully acknowledges support via USDA National Institute of Food and Agriculture, Hatch project number #1001480.

CONFLICT OF INTEREST

The author(s) declare no competing financial interests.

AUTHOR CONTRIBUTIONS

D.A. formed the interdisciplinary team and proposed to couple numerical optimization and plant physiology. D.A, M.E.G., and B.G. designed the research plan. M.E.G. designed the crop hydraulic model. B.G. implemented the model. B.G and T.Z.J. designed the optimization framework. T.Z.J. ran the simulations and processed the data. All authors improved the model, analyzed the data, and wrote the manuscript.

1 APPENDIX

PART A: DESCRIPTION OF THE PHYSIOLOGICAL MODEL AND NUMERICAL IMPLEMENTATION

The full plant model is constructed using various submodels as shown schematically in Figure .

WATER TRANSPORT SUBMODEL

Uptake of water by the root from the soil reaches the top of the canopy due to cohesive‐adhesion interactions, but the main energy source for this transport is the dryness of the atmosphere. Water travels from the soil to the leaf as a liquid. Subsequently, as a gas, it evaporates from the leaf (through the stomatal openings) to the surrounding environment. The evaporation rate depends on the leaf temperature, external relative humidity, air temperature, and boundary layer effects. This is termed environmental water demand. To fulfill this demand, the plant supplies liquid water to the leaf. This supply is driven by the potential difference of water between the soil and the leaf and is controlled by the hydraulic resistance of the plant.

Using a one‐dimensional representation of the plant hydraulic characteristics, as shown in Figure , the water supply, $J_{W,s}$, can be expressed as (Park, )[Image Omitted. See PDF]where K_plant is hydraulic conductance of the plant, K_soil is hydraulic conductance of the soil, ψ_leaf is the water potential at the leaf, and ψ_soil is the water potential in the bulk soil. Water potential is a combined effect of hydrostatic pressure, osmotic pressure, matric pressure, and gravitational pull. Osmotic pressure that depends on the presence of ions in the water is neglected in our model, as we consider the water as pure and free from any minerals. This is effectively the case for water in xylem and is typically an acceptable approximation in the soil.

The hydraulic conductance of the plant can be expressed as[Image Omitted. See PDF]where each component is the hydraulic conductance of the root, stem, and leaf, respectively.

The hydraulic conductance of the soil, K_soil, depends on the type of soil, the amount of water in the soil, and the relative occupancy of the root in the soil. The effect of these parameters is captured via the following equation (Cowan, ),[Image Omitted. See PDF]where k_sat, b, ψ_sat,_, and ψ_soil vary among the types of soil, and they represent saturated hydraulic conductivity, texture, the water potential of saturated soil, and water potential of the soil, respectively. The rest of the terms are used to capture the effect of the presence of root on the soil conductance. The symbols L, H_s, and r_root represent root length density of the absorbing root (length per soil volume), depth of the soil occupied by the root, and radius of the root.

The water potential of the soil, ψ_soil, can be expressed as a function of soil water content using an empirical equation presented by Campbell and Norman () as[Image Omitted. See PDF]where θ_sat is the saturated water content in the soil, and θ is current volumetric water content in the soil. In this model, soil water content would gradually deplete as plants fulfill the atmospheric water demand. The depletion of water due to evaporation of water from the soil is not considered here as it is typically small in closed canopies. Under irrigated conditions, based on the irrigation frequency, water is added to the soil until water content reaches θ_sat of the soil. For example, for irrigation frequency 7, the soil is fully saturated every 7 × 24 hr. However, in the case of drought conditions, water in the form of precipitation or irrigation could be “turned off.”

Soil water potential at the root (Equation 3), ψ_soil,root, can be evaluated from leaf water potential and water demand by the plants,[Image Omitted. See PDF]

Water demand is driven by the gradient of water vapor concentration between the leaf and the surrounding environment and is controlled by the stomatal conductance and air boundary layer conductance. It can be expressed as (Park, )[Image Omitted. See PDF]where P_vl and P_va represent water vapor pressure in the leaf and atmosphere, respectively, and P_a the atmospheric pressure. Water vapor pressures are evaluated using Tetens formula (Buck, ), $P_{v,i}={\text{RH}}_i{c}_0{e}^{\left(\frac{c_1{T}_i}{T_i+c_2}\right)}$, where RH is the relative humidity, c₀ = 0.617 kPa, c₁ = 17.38, and c₂ = 239°C. Generally, the leaf intercellular space is close to equilibrium with the cells having a relative humidity of greater than 99%, and thus, for each of calculation of evaporation, we consider the leaf to be fully saturated. g_st and g_blc are the stomatal conductance and boundary layer conductance to the water vapor transport, respectively.

Boundary layer conductance to water vapor, g_blc, depends on the atmospheric wind speed and the morphology as well as the orientation of the leaf. Wind speed and leaf dimension are designated as U_c, and d as in Campbell and Norman (). The conductance of water vapor through the air boundary layer on the leaf can be considered as forced convection and can be expressed via an empirical equation. Note that, here, the contribution from the free convection is neglected, as the ratio of dimensionless parameters Re²/Gr which reflects the forced convection/free convection is usually much greater than one. The empirical correlation among the dimensionless Reynolds number, Re, and Schmidt number, Sc, and the conductance can be calculated as,[Image Omitted. See PDF]where $\text{Re}=\frac{U_{\mathrm{c}}{d}_{\mathrm{e}}}{{\nu }_{\mathrm{a}}}$; $\text{Sc}=\frac{{\nu }_{\mathrm{a}}}{D_{\begin{array}{c}\text{WV}\hfill \\ \hfill \end{array}}}$; α = 0.644*1.4 is an empirical parameter; and d_e = 0.72 d, with d being the width of the maize leaf and 0.72 being used to find the equivalent parabola of the leaf where the wind is flowing in the width direction of the parabola. U_c, ν_a, and D_wv represent the wind speed on the top of the canopy, kinetic viscosity of air, and water vapor diffusivity in the air.

Wind speed can increase approximately logarithmically with distance above a plant canopy and is also influenced by the plants. The variation in wind speed can be described by[Image Omitted. See PDF]where 0.4 is related to the von Karman constant, H_c is the height of the plant, mH_c is the zero‐plane displacement, and nH_c is the roughness length. Generally, m is 0.7 and n is 0.1. U^* is termed the shearing or friction velocity and can be calculated from the wind speed U_m that is measured at height H_m from the ground as[Image Omitted. See PDF]

A small fraction of water that is absorbed from the soil is used by the plant for metabolism/growth, and <0.1% is used for photosynthesis.

CO₂ TRANSPORT AND NET PHOTOSYNTHESIS SUBMODEL

Along with water, the plant needs CO₂, sunlight, and enzymes for photosynthesis. From the environment, gaseous CO₂ diffuses into the leaf via stomata and then dissolves in water and diffuses to the cells where photosynthesis takes place. The consumption of CO₂ during photosynthesis depends on the sunlight and enzyme activity (plant cells desiccate before water becomes limiting to split in the photosynthesis).

The rate of gaseous CO₂ transport to the leaf is named as CO₂ supply. The supply is driven by the CO₂ concentration gradient between the atmosphere and the leaf intercellular space and is controlled by the conductance of stomata and the air boundary layer. This supply can be expressed as (Park, )[Image Omitted. See PDF]where β and χ are the ratios of CO₂ conductance and water vapor conductance through stomata and air boundary layer, respectively. β is the ratio of the molecular diffusivities of H₂O and CO₂, χ is power ¾ of β, and C_C,a and C_C,i are the concentration of CO₂ at the atmosphere and inside the intercellular space of the leaf.

The demand for atmospheric CO₂ depends on the supply of sunlight and the performance of the enzymes that control photosynthetic activity. The plant gets some CO₂ as a byproduct of metabolism or respiration activity in the mitochondria, and it lowers the atmospheric CO₂ demand.

For C4 plants, the electron transport to support CO₂ reduction occurs in mesophyll (C4 cycle) and bundle sheath (C3 cycle) cells. If the supply of sunlight is lowered compared with enzyme performance, which mainly occurs during the morning, sunset, or cloudy days, the photosynthetic rate can be expressed as (Caemmerer, )[Image Omitted. See PDF]where J_e,t is the total electron transport rate at leaf temperature, R_t is the rate of CO₂ production from respiration in the mesophyll and bundle sheath cell, and $x$ is a fraction of total electrons that are used by the mesophyll.

PEPCase, phosphoenolpyruvate carboxylase, and Rubisco are two enzymes that significantly control the photosynthesis activity in C4 plants. PEP (three‐carbon backbone) controls the activity of the mesophyll cell (it catalyzes the primary carboxylation in the mesophyll tissue that is close to the internal leaf atmosphere), and Rubisco controls activity in bundle sheath cell. In the case of no limitations on the supply of reductant to photosynthesis (higher light intensities), the photosynthetic demand can be expressed as (Caemmerer, )[Image Omitted. See PDF]where the top expression in the right‐hand side depends on the performance of PEPCase in the mesophyll cell, and the bottom expression depends on the Rubisco performance in the bundle sheath cell. g_C,bs is the bundle sheath conductance to CO₂, C_C,m is the concentration of CO₂ in the mesophyll cell (note that we assume that C_c,m = C_c,I, CO₂ concentration in intercellular space), R_m is mitochondrial respiration in the mesophyll at leaf temperature (i.e., CO₂ supply from the respiration of the mesophyll cell), R_t is the total mitochondrial respiration in the mesophyll and bundle sheath at leaf temperature, and $V_{{\text{RO}}_{max}}$ is the maximum Rubisco carboxylation rate.

V_PEP is the effective PEP carboxylation at leaf temperature. It depends on the availability of CO₂ and the regeneration of PEP and can be expressed as (Caemmerer, )[Image Omitted. See PDF]where the top expression in the right‐hand side is related to the carboxylation rate of PEP, expressed with the Michaelis–Menten equation. C_C,m is the CO₂ partial pressure in mesophyll, $V_{{\text{PEP}}_{max}}$ is the maximum PEP carboxylation rate at leaf temperature, and K_p is the Michaelis–Menten constant for PEP carboxylase for CO₂ at leaf temperature. Note that the Michaelis–Menten constant, K_p, refers to the concentration of CO₂ at which the reaction rate is half of $V_{{\text{PEP}}_{max}}$. The carboxylation rate can be decreased if there is not enough PEP, and that depends on the V_PEP,R, the PEP regeneration rate at leaf temperature.

The temperature‐dependent properties are evaluated using the following equations (Collatz, Ribas‐Carbo, & Berry, )[Image Omitted. See PDF][Image Omitted. See PDF]

[Image Omitted. See PDF]

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[Image Omitted. See PDF]where, A, B, C, J_a, and J_b are physiological parameters related to the carboxylation rate and electron transport rate. The subscript 25 in the symbols indicates the parameters at 25^◦C. Hourly J_e,25 can be expressed as (Caemmerer, )[Image Omitted. See PDF]where λ is the empirical curvature factor and $I=\text{PAR}\times f_{\text{PAR\_PSII}}$. f_{PAR_PSII} is the fraction of PAR that contributes to the photosystem II.

Using the photosynthesis rate (PR) of the above two limiting cases, the CO₂ demand can be expressed as (Caemmerer, )[Image Omitted. See PDF]

THE ENERGY BALANCE OF LEAF SUBMODEL

In the above submodels, many of the parameters related to leaves, for instance, water vapor pressure, and enzyme activities, depend on the leaf temperature. Leaf temperature can be evaluated by using first principles in so‐called “big leaf models” (Campbell & Norman, ). Several assumptions are considered in this model: The leaf is flat and perpendicular to the incident sunlight; leaf does not store any energy, and there is negligible heat generation due to metabolic activity in the leaf. Considering a leaf that is at steady state, the energy balance equation of a leaf can be expressed as (Campbell & Norman, )[Image Omitted. See PDF]where the terms are energy input by solar irradiation and the surrounding irradiation, cooling by leaf irradiation, convective/conductive cooling by the air/temperature gradient, and heat loss accompanying water evaporation. In Equation (23), a is the absorptance of the leaf, r is the reflectance, S is the solar irradiation, a_IR is the absorptance of leaf for thermal infrared radiation, L_vap is the latent heat of vaporization of water, h_c is the convective heat transfer coefficient, and g_hbc is the air boundary conductance to heat transfer.

The boundary layer conductance depends on leaf morphology and wind speed and can be expressed via empirical relationships of dimensionless parameters Reynolds number, Re, and Prandtl number, Pr. It can be expressed as[Image Omitted. See PDF]where $\text{Re}=\frac{U_{\mathrm{c}}{d}_{\mathrm{e}}}{{\nu }_{\mathrm{a}}}$; $Pr=\frac{{\nu }_{\mathrm{a}}}{D_{\mathrm{H}}}$; β = 0.644*1.4 is an empirical parameter; d_e = 0.72 d, with d being the width of maize leaf and 0.72 being used to find the equivalent parabola of the leaf where the wind is flowing in the width direction of the parabola. U_c, ν_a, and D_H represent the wind speed on the top of the canopy, kinetic viscosity of air, and thermal diffusivity in the air. The effect of the temporal variation of soil is not explicitly included in Equation (23). Instead, the effect is implemented using the FAO‐56 algorithm, as in Allen, Pereira, Raes, and Smith ().

STOMATAL CONDUCTANCE SUBMODEL

In the pathway of the supply of CO₂ Equation (10) from the environment and demand of H₂O (Equation 6) to the environment, stomatal conductance is the most significant factor. Stomatal conductance is a very complex parameter that is affected by the environment, plant physiology, and heredity.

At least 35 empirical models have been proposed to capture the complex relationship between stomatal conductance and various factors (Damour, Simonneau, Cochard, & Urban, ; Dong et al., ; Dunbabin, McDermott, & Bengough, ; Guswa, ; Huntingford et al., ; Jensen et al., ; Menge, Ballantyne, & Weitz, ; Sellers, Mintz, Sud, & Dalcher, ; Zwieniecki, Stone, Leigh, Boyce, & Holbrook, ). Such factors include environmental factors, for example, solar radiation, soil water content, humidity, and wind speed, and physiological factors, for example, leaf water potential, root water potential, and hydraulic root conductance. Few models explicitly include the plant physiological influences on the stomatal conductance apart from entirely empirical functions. Here, we propose a model which is developed based on the sigmoidal response of the stomatal conductance with respect to the leaf water potential (Brodribb & Holbrook, ). The main concept of this model is shown in Figure . Here, the stomatal conductance will start decreasing when leaf water potential touches the threshold potential, which depends on the plant variety/genotype. The sensitivity of the decrease (closure) of stomatal conductance to water status is controlled by the two sensitivity terms S_l (related to leaf water potential) and S_r (related to root water potential). The model is expressed as,[Image Omitted. See PDF]where the environmental response on the stomatal conductance is implicitly influenced by J_C,d, and ψ_leaf. ψ_th is the threshold bulk leaf water potential at stomatal closure, S_r is the slope of the relationship between stomatal conductance and root water potential, and ψ_root. $g_1$ and $g_2$ are plant physiological properties related to photosynthesis. Z is a parameter to make the exponent dimensionless. Thus, the model can represent stomatal responses to light and photosynthesis (g1, g2), leaf water status (ψ_leaf), and chemical signaling from drying roots (ψ_root); the inclusion of ψ_leaf allows the stomata to close in response to both atmospheric demand and supply constraints. Alternative models typically independently represent stomatal closure in response to empirical relationships of the closure relative to soil water content and atmospheric evaporative demand.

METHOD TO EVALUATE NET PHOTOSYNTHESIS AND WATER USE

Figure shows the schematic of the modeling concept, and Figure shows the flowchart of the model implementation. For the input weather conditions, soil and agronomic/management practices the total integrated net photosynthesis and water use (transpiration) can be evaluated iteratively by satisfying Equations 6 10, 23, and 25. A plant is considered dead, and net photosynthesis is zero if the plant experiences a permanent wilting condition or permanent temperature damage.

PART B: WHY WERE GROUP 6 PLANTS NOT CONSIDERED?

Table 2 contains the values of parameters of a typical plant in group 6. The typical plant in group 6 has higher S_r than S_l, which makes root water potential the dominating factor in the stomatal closure window of ψ_leaf, where upper end is ψ_th when stomata start closing and the lower end is the value of ψ_leaf when stomata reach g_min. In well‐irrigated conditions, root water potential is always higher than that in drought condition. In both cases, leaf water potential is low because of the very low value of plant hydraulic conductance. As drought soil experiences lower root water potential, for the similar values of leaf water potential, the stomata closure window is smaller than that in the case of well‐watered soil. Due to higher stomata closure window, in some weather conditions, the water potential at fully closed stomata (g_min) crosses the leaf wilting potential, that is, even fully closed stomata cannot prevent plant death. This kind of plant is unlikely to exist in nature. Therefore, we did not consider this group in our analysis. See Figure