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. 2022 May 6;111(1):38–53. doi: 10.1111/tpj.15774

Multi‐objective optimization of root phenotypes for nutrient capture using evolutionary algorithms

Harini Rangarajan 1, David Hadka 2, Patrick Reed 3, Jonathan P Lynch 1,✉
PMCID: PMC9544003  PMID: 35426959

SUMMARY

Root phenotypes are avenues to the development of crop cultivars with improved nutrient capture, which is an important goal for global agriculture. The fitness landscape of root phenotypes is highly complex and multidimensional. It is difficult to predict which combinations of traits (phene states) will create the best performing integrated phenotypes in various environments. Brute force methods to map the fitness landscape by simulating millions of phenotypes in multiple environments are computationally challenging. Evolutionary optimization algorithms may provide more efficient avenues to explore high dimensional domains such as the root phenotypic space. We coupled the three‐dimensional functional–structural plant model, SimRoot, to the Borg Multi‐Objective Evolutionary Algorithm (MOEA) and the evolutionary search over several generations facilitated the identification of optimal root phenotypes balancing trade‐offs across nutrient uptake, biomass accumulation, and root carbon costs in environments varying in nutrient availability. Our results show that several combinations of root phenes generate optimal integrated phenotypes where performance in one objective comes at the cost of reduced performance in one or more of the remaining objectives, and such combinations differed for mobile and non‐mobile nutrients and for maize (a monocot) and bean (a dicot). Functional–structural plant models can be used with multi‐objective optimization to identify optimal root phenotypes under various environments, including future climate scenarios, which will be useful in developing the more resilient, efficient crops urgently needed in global agriculture.

Keywords: food security, root system architecture, phene, phene interactions, multi‐objective optimization

Significance Statement

Analyzing and understanding the complexity of the phenome is a grand challenge in biology. The fitness impacts of a specific trait or ‘phene state’ depends not only on the environment but also on other traits expressed in that organism. By linking a functional structural plant model to a multi‐objective evolutionary algorithm, we are able to identify optimal root phenotypes in different nutrient scenarios. Our results demonstrate that this approach has utility in matching root phenotypes to their environments.

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INTRODUCTION

Root system architecture plays key roles in a range of processes from anchorage to nutrient and water acquisition (Lynch, 1995, 2021; Voss‐Fels et al., 2018). Increased nutrient acquisition can improve yields in the low fertility soils characteristic of low‐input agroecosystems and increase fertilizer use efficiency in high‐input agroecosystems (Lynch, 2019). However, an important obstacle to deploying root phenotypes in crop breeding is our limited understanding of their fitness landscape, i.e., the utility of specific root phenotypes in specific environments (Lynch, 2019; Schneider & Lynch, 2020).

Root phenotypes are comprised of multiple phenes (phenes are elementary units of the phenotype, and they are related to phenotypes as genes are to genotypes; Lynch & Brown, 2012; Rangarajan & Lynch, 2021; York et al., 2013) that exist in multiple states, which may be beneficial in specific scenarios. Fitness trade‐offs for contrasting soil resources, and between abiotic and biotic constraints, are important (Lynch, 2019). Dynamic constraints such as carbon availability further increase the complexity of the system (Rangarajan et al., 2018). For example, P‐deficient plants have smaller leaves and slower leaf appearance resulting in reduced sink strength of the shoot, thereby increasing relative allocation of carbon to the root system, whereas N deficiency slows growth, sometime severely, due to reduction in photosynthetic efficiency, both engendering carbon limitations, albeit in different ways (Lynch et al., 1991; Sinclair & Horie, 1989). Identifying optimal integrated phenotypes is a highly complex, non‐trivial challenge.

Considering multiple states for each phene, phene synergisms and antagonisms, acquisition of multiple nutrients simultaneously, multiple soil types, multiple precipitation regimes etc., the number of relevant scenarios is exceedingly large, and is well beyond the capacity of empirical experimentation. Moreover, when evaluating the functional benefits of phene interactions, there are conflicting objectives (e.g., maximize shoot biomass production, maximize resource capture, minimize carbon cost, etc.). Therefore, the challenge of mapping and understanding the fitness landscape for root phenotypes is a hugely complex and challenging non‐linear problem. In a recent study with three root phenes in contrasting and extreme phene states combined factorially, interactions among phenes in combination with trade‐offs due to carbon limitations resulted in several distinct root architectures with varied fitness in environments varying in nutrient availability. However, there is no single optimal architectural phenotype, and there exist multiple co‐optimal root architectural phenotypes for a given environment (Rangarajan et al., 2018). These in silico results were indirectly supported by the observation that multiple integrated root phenotypes are associated with improved drought tolerance in maize (Klein et al., 2020).

Simulating phenotypes representing various combinations of underlying phenes, evaluating their performance and optimizing for various objectives, some of which are conflicting, in various environments, is difficult and computationally intensive. One approach to identify optimal root phenotypes in a varying nutrient landscape is via Multi‐Objective Evolutionary Algorithms (MOEA; Coello Coello et al., 2007). Multi‐objective algorithms were chosen over other optimization techniques because there are multiple objectives to optimize, and most of the decision variables (i.e., input parameters) are continuous, so by discretizing the decision space, there are an extremely large number of model configurations to evaluate. Inherent trade‐offs exist between components of fitness, which limits the set of potential phenotypes (Ho et al., 2005); therefore, the integrated phene space is discontinuous. Evolutionary algorithms in multi‐objective search and optimization are effective in their ability to handle complex problems, involving features such as discontinuities, multimodality, disjoint feasible spaces, and noisy function evaluations (Fonseca & Fleming, 1995; Reed et al., 2013). Multi‐objective evolutionary algorithm frameworks, particularly Borg (Hadka & Reed, 2013), have been shown to give better solutions than other evolutionary algorithms and random evaluations in several real‐world optimization applications. The premise of multi‐objective optimization is that the best phenotype for one task is usually not the best for other tasks—resulting in a fitness trade‐off. Trade‐offs occur when the benefit of one trait comes at the cost of allocating resources to a different trait (Kimball et al., 2013). Biomass is the result of trade‐offs between uptake of nutrients, which require the deployment of strategies that are conflicting in functionality. Trade‐offs also exist among different phenes in terms of function as well as carbon investment required. Because of these trade‐offs, different strategies are adapted by different phenotypes to obtain comparable shoot biomass. Addressing the optimization as a single objective problem, i.e., evaluation of the phenotypes based solely on their shoot biomass, will result in a single optimal solution while disregarding a wealth of information regarding comparable phenotypes. The result of the multi‐objective optimization is a set of non‐dominated solutions (Pareto optimal solutions), which are points on the Pareto front (Coello Coello et al., 2007; Noor & Milo, 2012; Shoval et al., 2012). This set comprises phenotypes whose performance in one objective cannot be improved without reducing performance in the other objectives.

SimRoot, a functional–structural plant model (Lynch et al., 1997) has been used extensively for elucidating the functional value of one or more phenes, and to analyze phene interactions (e.g., Ajmera et al., 2022; Rangarajan et al., 2018). In this study, we use the Borg MOEA with SimRoot to identify optimal root phenotypes of common bean (Phaseolus vulgaris) and maize (Zea mays), representing a dicot and a monocot species that are primary global food security crops. The Borg MOEA searches the decision space constrained by a range of root phenes while optimizing for several objectives including maximizing P uptake, nitrate uptake, shoot biomass, and rooting depth while minimizing carbon investment in root construction and maintenance including root respiration. Numerous combinations of root phenes could generate integrated phenotypes and such combinations differ between monocots and dicots, and among taxa within these groups (Lynch, 2019). Using SimRoot with the Borg MOEA, we identify a diverse suite of optimal integrated common bean and maize root phenotypes, which have optimal P and nitrate uptake, representing mobile (nitrate) and immobile (P) soil resources, under a dynamic constraint imposed by carbohydrate availability.

RESULTS

The objective space

In total, 2700 maize solutions and 2400 bean solutions corresponding to optimal phenotypes in environments varying in nitrate and P availability were obtained. The Pareto solutions were obtained in the multidimensional objective function space (shoot biomass, P uptake, N uptake, carbon cost, root respiration for the bean root system and root length at depth additionally for the maize root system). The Pareto front is mapped on to the two‐dimensional self‐organizing map (SOM) (Kohonen, 1998), according to the scaled objective function values, where trade‐offs are successfully visualized. The performances of the various phenotypes in different objectives in the optimal solution set of bean and maize root system in a region with low N + P is visualized as a SOM heatmap in Figure 1(a,c) respectively. The complete Pareto set obtained from the bean and maize optimization routines is shown in Figure S1.

Figure 1.

Figure 1

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Self‐organizing map (SOM) heatmap of performance of root phenotypes in different objectives in a region with low N + P.

The objectives are clustered by SOM. Each cluster (node) in the heatmap has phenotypes with similar performance in all objectives. (a, c) Average value of the objective in bean and maize root systems respectively in that node. (b, d) Relative performances of phenotypes in all the objectives in bean and maize root systems respectively in that node. (a, b) Node 9; (c, d) node 7. Regions with low carbon cost and low respiration, low P uptake, N uptake, and low biomass. (c, d) Node 9. Phenotypes with very good N uptake or P uptake but shoot biomass depended upon the carbon invested in the root system. (a, b) Node 4; (c, d) node 3. Phenotypes with optimal biomass had intermediate performances in all other objectives. [Colour figure can be viewed at wileyonlinelibrary.com]

Regions with the greatest shoot biomass corresponded to regions with greatest nutrient availability; however, not all the phenotypes that evolved under the greatest nutrient availability had a high shoot biomass. The regions with low carbon cost and low respiration typically had low P uptake, nitrate uptake and so, low shoot biomass also (Figure 1; Figure S1). Phenotypes with good shoot biomass varied in carbon costs. Regions with optimal shoot biomass were regions that had intermediate performances in all other objectives (Figure 1; Figure S1).

Different phenotypes had different performances in different objectives in the same region of the nitrate‐P (NP) landscape. Trade‐offs in performance of different objectives were observed and were specific to distinct NP regions (Figure 2). The trade‐offs in performance in the objective was specific to specific NP regions. For example, phenotypes optimizing for greater P uptake in a low P environment were also the ones that had the greatest shoot biomass (Figure 2: Phenotype a). However, phenotypes optimized for P uptake in a low N environment had less than optimal shoot biomass for that region of the NP landscape (Figure 2: Phenotype b). The maize root optimization routine included an objective to find optimal phenotypes that had the greatest root length in deeper soil strata, i.e., maximize root length at greater depth. Many phenotypes with the greatest root length in deeper soil strata were also the phenotypes that had good nitrate uptake and consequently good shoot biomass (Figure S1(b) region 3). There were also phenotypes that had good root length but were not as efficient in accumulating shoot biomass (Figure 2).

Figure 2.

Figure 2

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Trade‐offs in performance of different objectives in optimal maize root phenotypes: phenotypes optimizing for greater N uptake in a low N environment were also the ones that had the greatest biomass (a), while those optimizing for greater P uptake in a low N environment had lower biomass (b).

Phenotypes with the greatest root length in deeper soil strata were also the phenotypes that had good N uptake and consequently good biomass (g). There were also phenotypes that had good root length but were not as efficient in accumulating biomass (f). The fan plot shows the relative performance of the phenotype in different objectives. Primary root is in black; seminal roots in red; nodal roots in green. Units shown are cm. [Colour figure can be viewed at wileyonlinelibrary.com]

Different phenotypes had similar performances in at least one of the objectives in the same region of the NP landscape, i.e., multiple optimal phenotypes existed for an objective (Figures S2 and S3). In this study we focus on optimal phenotypes for shoot biomass in bean and maize irrespective of how they performed in the other objectives, in specific regions of the NP landscape corresponding to low P, low N, and low N + P.

Preliminary investigation of the optimal phenotypes suggested that not all combinations of states of different phenes were represented in the final optimal solution set. The phene states of the constituent phenes represented in the optimized phenotypes had very skewed distribution (Figure 3). For better interpretation of the characteristics of the optimal solutions, the phenotypes were analyzed at the root class levels. The phene states of different root classes represented in the bean and maize optimal root phenotypes are presented in Figures S4–S7.

Figure 3.

Figure 3

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Distribution of phene states in optimal root phenotypes of bean and maize in low N, low P and low N + P regions of the landscape.

(a) Distribution of phene states of bean root phenes in optimal bean root phenotypes in low N, low P, and low N + P regions of the landscape.

(b) Distribution of phene states of maize root phenes in optimal maize root phenotypes in low N, low P, and low N + P regions of the landscape. Optimal root phenotypes under low P had the greatest lateral root branching density (LRBD), shallowest angles, greatest number of roots, and whorl occupancy while those under low N had lower LRBD. The outliers are not interpreted in the traditional sense and correspond to unique phene states of some of the optimal phenotypes. #, number of roots; BW, basal root; Dia, diameter; HBR, hypocotyl‐borne root; NR, nodal root; PR, primary root; SR, seminal root. [Colour figure can be viewed at wileyonlinelibrary.com]

The optimal phenotypes that evolved in different regions of the NP landscape were based on certain root class‐specific phenotypes. None of the phenotypes had the maximum potential value for all the phenes even under non‐limiting nutrient conditions, i.e., huge root systems were not found in the optimal set. There were differences in root class phenotypes in different regions of the NP landscape (Figure 3). The characteristics of root classes emerging later in development depended upon the already emerged phenotype (Figure 4) as well as nutrient availability in a particular region of the NP landscape (Figure 5).

Figure 4.

Figure 4

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Maize root classes emerging later in development depended upon the already emerged phenotype.

Different seminal root (SR) phenotypes found in the optimal maize phenotypes. SR phenotype is defined by the number of root axes, and their angle, diameter and lateral root branching density (LRBD). Phenotypes differing in all of these phenes were found among the optimal solutions in maize. Some optimal phenotypes had no SR whereas some had many SR. Under low P, phenotypes had fewer SR, which were highly branched and had shallow growth angles. Under low N, phenotypes had deep angled SR, with very few lateral roots. Phenotypes under low P had shallow SR, while those under low N had deep angles and those under low N + P had intermediate SR angles. Greatest LRBD in SR was found in regions with low P. The SR characteristics were dependent on the primary root (PR). For example, if the PR diameter was >2 mm, there were more constraints on the possible phene states of SR phenes; there was a constraint on SR LRBD in that high LRBD SR were not found when the PR diameter was large. Nodes shaded in gray contain the SR phenotypes that are seen when PR have larger diameter. [Colour figure can be viewed at wileyonlinelibrary.com]

Figure 5.

Figure 5

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Maize root classes emerging later in development depended upon nutrient availability in a particular region of the NP landscape.

Phenotypes with similar PR ideotypes and different SR and NR ideotypes in optimal maize phenotypes in regions with low P, low N, and low N + P. Heatmap showing the phene states of different phenes of optimal maize root system under low P, low N, and low N + P (a) and the respective phenotypes are visualized in (b). The distribution of the variables are represented color coded. White represents the absence of the particular class of roots and traits associated with that root class. NR1–4, nodal roots 1–4; PR, primary root; SR, seminal roots. Lower values of the variable are in blue changing to yellow and red with increase in numerical value of the particular trait. #, number of roots; Dia, axial root diameter; LRBD, lateral root branching density. PR is in black; SR in red; NR in green. Units shown are cm. [Colour figure can be viewed at wileyonlinelibrary.com]

Phenotypic space/morphospace

Preliminary investigation of the optimal phenotypes suggested that not all combinations of states of different phenes were represented in the final optimal solution set. The states of the constituent phenes represented in the optimized phenotypes had a very skewed distribution. For better interpretation of the characteristics of the optimal solutions, the phenotypes were analyzed at the level of root class.

Primary root

The primary root (PR) phenotype in bean and maize is defined by diameter and lateral root branching density (LRBD). Phenotypes with PR differing in both diameter and LRBD were found among the optimal solutions of both bean and maize (Figure S4a,b). None of the optimal phenotypes had high values for both diameter and LRBD in bean or maize. Some phenotypes had very large diameters and some very high LRBDs (Figure S4a,b). Phenotypes with large diameter PR typically had low shoot biomass and low carbon cost and lower respiration in bean as well as maize. Phenotypes with very high LRBD were seen in phenotypes optimal under low P in bean and in maize root systems. Roots under low P had smaller PR diameter than under low N or low N + P (Figure 3a,b).

Hypocotyl‐borne roots in bean

The hypocotyl‐borne root (HBR) phenotype is defined by the number of root axes, their diameter, and LRBD, and phenotypes differing in all three of these phenes were found among the optimal solutions in bean (Figure S5). Phenotypes with more HBR as well as greater LRBD of HBR were found in optimal phenotypes evolved under low P (Figure 3a). Some phenotypes did not have any HBR and were typically found in regions low in N and under very low P (Figure S5).

Seminal roots in maize

The seminal root (SR) phenotype is defined by the number of root axes, and their angle, diameter, and LRBD. Phenotypes differing in all of these phenes were found among the optimal solutions in maize (Figure 4). Some optimal phenotypes had no SR whereas some had many SR. Under low P, phenotypes had fewer SR, which were highly branched and had shallow growth angles. Under low N, phenotypes had deep angled SR, with very few lateral roots. Phenotypes under low P had shallow SR, while those under low N had deep angles and those under low N + P had intermediate SR angles. The greatest LRBD in SR was found in regions with low P (Figure 3b). The SR characteristics were dependent on the PR. For example, if the PR diameter was >2 mm, there were more constraints on the possible phene states of SR phenes; there was a constraint on LRBD SR in that high LRBD SR were not found when PR diameter was large (Figure 4).

Basal roots in bean

The basal root (BR) phenotype is defined by the number of root axes, and their angle, diameter and LRBD as well as the BR whorl number. Small diameter, highly branched, shallow BR were found almost exclusively in low P regions as well as in low N + P (Figure 3a). BR phenotypes in low N are distinctly different from those expressed in low P conditions and typically had more BR with fewer lateral roots (Figure 3a) and had a wide range of root growth angles. The BR phenotypes found in the optimal bean phenotypes are presented in Figure S6.

Nodal roots in maize

The nodal root (NR) phenotype is defined by the number of root axes, and their angle and diameter as well as time of emergence. Optimal phenotypes under low P had fewer NR while those under low N had more NR (Figure 3b). Phenotypes with the greatest NR LRBD were found in the low P region (Figure 3b). The NR phenotypes found in the optimal maize phenotypes are presented in Figure S7.

Integrated phenotypes

The optimal phenotypes that evolved in different regions of the NP landscape were based on certain root class specific phenotypes. None of the phenotypes had the maximum potential value for all the phenes even under non‐limiting nutrient conditions, i.e., huge root systems were not found in the optimal set. There were differences in root class phenotypes in different regions of the NP landscape. Optimal phenotypes under low P had the greatest LRBD, shallowest angles, greatest number of roots (Figure 5a,b), and whorl occupancy.

The characteristics of root classes emerging later in development depended upon the already emerged phenotype as well as nutrient availability in a particular region of the NP landscape. For example, optimal maize roots with very highly branched PR were found in regions with suboptimal P, suboptimal N as well as suboptimal N + P regions but with different states of phenes of the SR and NR (Figure 5). While phenotypes in suboptimal P had no SR (Figure 5: Phenotype a), those under suboptimal N had a large number of SR with steep growth angles (Figure 5: Phenotype b) and when both P and N were suboptimal, the phenotype with highly branched PR had shallow‐angled SR (Figure 5: Phenotype c).

Phenotypes under low P had greater BR LRBD and more HBR with shallow angled BR resulted in optimal phenotypes in low P (Figure 6a). Larger diameter, greater whorl occupancy with few roots at each whorl, varying number of BR with varying angles and very small LRBD as compared with optimal phenotypes under low P were characteristic of optimal bean root phenotypes under low N (Figure 6a). Optimal bean phenotypes under low N + P conditions had phenotypes with few or medium number of lateral roots, intermediate growth angles, and few or no HBR (Figure 6a).

Figure 6.

Figure 6

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Optimal bean and maize root phenotypes in low P, low N and low N + P regions of the landscape.

(a) Heatmap showing the phene states of different phenes of optimal bean root systems under low P (Low P1, Low P2, Low P3), low N (Low N1, Low N2, Low N3), and low N + P (Low N + P1, Low N + P2, Low N + P3), and the respective phenotypes are visualized. PR is in black; basal roots in red; hypocotyl‐borne roots in green.

(b) Heatmap showing the phene states of different phenes of optimal maize root systems under low P (Low P1, Low P2, LowP3), low N (Low N1, Low N2, Low N3), and low N + P (Low N + P1, Low N + P2, Low N + P3), and the respective phenotypes are visualized. The distribution of the variables are represented color coded. White represents the absence of the particular class of roots and traits associated with that root class. NR1–4, nodal roots 1–4; PR, primary root; SR, seminal roots. Lower values of the variable are in blue changing to yellow and red with increase in numerical value of the particular trait. #, number of roots; Dia, axial root diameter; LRBD, lateral root branching density. PR is in black; SR in red; NR in green. Low N1: More SR with intermediate branching and very few deep NR with very low LRBD. Low N2: SR with very few branches and many NR with very low LRBD, some NR with deep angles and some with shallow angles. Low N3: more SR with few LR branches and few NR with greater LRBD. Units shown are cm. [Colour figure can be viewed at wileyonlinelibrary.com]

Maize root phenotypes in low P had highly branched PR with no SR and few NR (Figure 6b), or PR with very low LRBD, no SR, and highly branched NR (Figure 6b) or PR with very few branches and highly branched shallow SR with few NR with very low LRBD (Figure 6b). While phenotypes with low P tended towards very high LRBD obtained in terms of high LRBD in PR, SR, or NR, those under low N had different phenotypes varying specifically only in SR LRBD and number of NR or NR LRBD and number of NR (Figure 6b). Under low N + P, varying number of SR and SR LRBD as well as varying NR number and NR LRBD with intermediate root growth angles were found among the optimal maize phenotypes (Figure 6b).

Varying all the parameters to perform sensitivity analysis would be computationally very expensive. So, we conducted a sensitivity analysis changing the number of NR in two distinct maize optimal phenotypes under low N (Low N1, Low N2) and analyzed the performance of the phenotypes in various objectives and found that the shoot biomass of several phenotypes with different numbers of NR were comparable, while the performance in other objectives varied. This suggests that when the state of a single phene is varied, the states a particular phene could occupy to result in optimal shoot biomass is not a single unique value (Figure 7c), but rather a range of values, as long as roots are allocated to regions with greater resource availability (Figure 7a,b) and there is balance in the trade‐offs in carbon costs and nutrient acquisition (Figure 7d,e).

Figure 7.

Figure 7

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Senstivity analysis of optimal maize root phenotypes in low N region of the landscape.

(a) Root length distribution of maize root phenotypes Low N1, Low N2, and Low N3. Color scale ranges from blue to red with blue being low values of root length. Regions in white depict absence of that root class in the phenotype. NR, nodal root; PR, primary root; SR, seminal root.

(b) N availability in the soil profile at 20 and 40 days.

(c) Performance of maize phenotypes Low N1, Low N2, and Low N3 with change in number of NR.

(d) Performance of maize phenotype Low N1 in different objectives with change in NR number. Performance of maize phenotype Low N2. [Colour figure can be viewed at wileyonlinelibrary.com]

DISCUSSION

Several functional structural models exist that simulate root growth, architecture (Dunbabin et al., 2013), and function, including SimRoot (and open source version OpenSimRoot) (Lynch et al., 1997; Postma et al., 2014), ArchiSimple (Pagès et al., 2014), RootTyp (Pagès et al., 2004), ROOTMAP (Diggle, 1988), SPACSYS (Wu et al., 2007), R‐SWMS (Javaux et al., 2008), RootBox, etc. (Leitner et al., 2010). The aim of this study was to identify optimal root phenotypes based on their constituent phenes. SimRoot has been shown to be useful in studies of phene utility, is parametrized based on empirical data and tracks carbon availability and costs making it convenient to implement a cost–benefit attribute that can be subjected to an optimization algorithm. In this study, we used SimRoot, a functional–structural plant model FSPM with Borg, an MOEA to identify optimal maize and bean root phenotypes in environments with varying availability of N and P. Using the Borg‐SimRoot framework, we identified optimal root phenotypes under varying levels of N and P in bean and maize. N and P are primary limitations to plant growth in terrestrial environments, and provide an interesting contrast in that N (as nitrate) is highly mobile in soil water whereas P is highly immobile in soil. These two resources therefore represent two broad classes of resources, i.e., mobile (including water and nutrients soluble in water such as nitrate, sulfate, Ca, Mg, silicate) and immobile (remaining nutrients). We analyzed root phenotypes of two species, maize and common bean, representing a monocot and a dicot root architecture. The main difference between dicot and monocot root systems is that in dicots new roots emerge from already existing roots, whereas in monocots NR continually emerge over time from shoot nodes near or above the soil surface. Dicot roots also thicken with time because of secondary growth, whereas monocots do not. Similarities and differences were seen among the optimal phenotypes in the two species. While the diameter of all root classes in both species were optimized towards thinner diameters, states of the other phenes varied based on the limiting nutrient. Some strategies for optimal nutrient uptake were similar while others differed between bean and maize. Multiple phenotypes with similar shoot biomass were seen in each region of the NP landscape. The optimal phenotypes were distinct based on trade‐offs among root classes for optimum nutrient acquisition to maximize shoot biomass while being economical in terms of carbon invested in the root system.

SimRoot is an FSPM that considers the dynamic feedbacks between structure and function, spatial and temporal heterogeneity in resource distribution, competition among root axes for internal and external resources, and costs and benefits of different root phene states and growth strategies (Lynch et al., 1997). SimRoot and its successor OpenSimRoot (Postma et al., 2017) have been extensively used to evaluate the utility of root phenotypes, estimate processes such as competition for soil resources within and among neighboring plants, and discover new phenotypes (Benes et al., 2020). Few studies have attempted to optimize root phenotypes, typically using simple representations of root structure and function and relatively few parameters (Dunbabin et al., 2003; Ho et al., 2004) and fewer studies have used evolutionary algorithms towards attaining this goal. Evolutionary algorithms and plant structural models have been used to explore multicriteria fitness landscapes for shoots (Niklas, 1994). A study by Renton and Poot (2014) used an evolutionary optimization algorithm to simulate the evolution of water foraging strategies using a simple representation of the dynamic root structure. The complexity of both the root phenotype and the soil environment, the large number of parameters involved and their dynamic nature make exploring all possible parameter combinations to identify optimal phenotypes a non‐trivial computational challenge (Lynch & Brown, 2012; Renton & Poot, 2014).

Optimal root phenotypes in low P

Phenotypes with maximum biomass under low P had the greatest LRBD in both maize and bean. Studies have shown that greater LRBD and more axial roots are independently beneficial for P uptake (Lynch, 2019). However, the optimal number of axial roots depends on the LRBD (Rangarajan et al., 2018). The phenotypes optimized under low P prioritized greater LRBD over production of more axial roots in maize as well as bean resulting in root phenotypes with greater soil exploitation, a requirement for the uptake of immobile soil resources such as P. Bean had shallow BR with a very narrow range of growth angles, which is in agreement with several studies that show that topsoil foraging is beneficial for P uptake (Lynch, 2019). The different alternate bean root phenotypes selected as optimal were variations of the same phenotype with occupancy at different whorls. BR emerge around the same time (Basu et al., 2007; Miguel et al., 2013) and as the states of the other BR phenes were similar except for the whorl position in several of the phenotypes, these phenotypes were not very distinct from each other, resulting in fewer distinct phenotypes under low P. Maize did not have very shallow root growth angles. Like other monocots, maize continually forms NR, which pass through topsoil as they descend to deeper soil strata and so are not dependent on an exclusively shallow angled root class for topsoil exploration, unlike dicots. In the case of maize, temporal variation in emergence of different classes of roots (Hoppe et al., 1986) results in more distinct phenotypes for P uptake as compared with bean. At very low P, the optimal maize phenotypes had a highly branched PR with an absence of SR. The large carbon cost imposed by a highly branched PR and absence of emerging SR for the next few days enabled better PR growth, which subsequently supported production of more NR. In bean, the BR emerge soon after germination, forming the scaffold of subsequent lateral roots. However, HBR emerge much later and are restricted in their root growth angle, growing almost horizontally, exploring almost exclusively the topsoil (Miller et al., 2003) unlike NR in maize, which have a greater range of growth angles (Trachsel et al., 2013). HBR are therefore well‐represented in the optimal phenotypes under low P but not under low N. HBR with greater plagiogravitropism in dicots such as those in cowpea (Burridge et al., 2016) could enable more varied phenotypes under low P similar to those seen in maize. The effects of other complementary traits for P acquisition such as root hairs, colonization by mycorrhiza, etc. (Galindo‐Castañeda et al., 2018; Hochholdinger, 2016) have not been included in this study and could certainly have a significant influence in determining optimal number of axial and LRBD.

Optimal root phenotypes in low N

Phenotypes evolving in a low N environment have to optimize against another level of complexity as compared with those in low P because nitrate availability varies greatly in time and space. Nitrate has greater mobility than P and competition for mobile resources is much greater than for immobile resources (Lynch, 2019). The greater number of constraints for N uptake results in a greater number of distinct optimal phenotypes under low N than under low P in both maize and bean. It is well established that the number of optimal phenotypes increases in proportion to the number of biological tasks that must be simultaneously performed (Niklas, 1997). Optimal phenotypes under N‐limiting conditions had steep root growth angles or had a wide range of growth angles and low LRBD in both maize and bean. The utility of low LRBD and steep root growth angles for N uptake under low N conditions is well established (Lynch, 2019). We found that optimal phenotypes under low N were those that were able to place roots where nutrient availability was greatest (Dathe et al., 2016) while being economical in carbon investment. By investing in axial roots with low LRBD, the optimal phenotype is able to reduce carbon cost, while the number and angle of axial roots at different nodes/whorls result in a wide range of angles allowing greater soil exploration by optimally placing roots in regions with greater nutrient availability. The performance of the optimal phenotypes was not sensitive to root growth angles as long as the angles were not too deep as roots with very deep angles resulted in competition between roots (Dathe et al., 2016; Ge et al., 2000; Rubio et al., 2003). The emergence of roots at different nodes sequentially over time enabled maize to adopt other strategies of optimizing nitrate uptake. One strategy was to develop a deep SR system with an optimal number of branches, such that the benefit of having more branches outweighs the effect of competition, enabling early vigorous root growth with deep soil exploration. SR are known to be important for seedling vigor during early development (Hochholdinger et al., 2018, Perkins & Lynch, 2021). In wheat, by the time the NR appear, the SR system was found to be up to 40 cm deep in the soil. Increasing root length contributed by SR is thought to increase water extraction from deeper soil layers (Richards, 2008). An added advantage of vigor during early development is that greater root and shoot development earlier during the season could synchronize with availability of N in the topsoil while reducing loss of N, particularly in soils prone to leaching. The phenotypes in the optimal set varied in the number of crown roots. While previous studies have shown that fewer crown roots are efficient for N capture (Saengwilai et al., 2014), our study shows that the optimal NR number depends on LRBD of NR as well as the branching frequency and number of SR. With an increased number of NR, the length of the lateral roots is reduced; however, the resulting phenotypes will still have comparable biomass as long as roots coincide with regions of high nutrient availability in time and space and, trade‐offs in the different root classes do not result in phenotypes with vastly different total carbon cost or resource acquisition.

Optimal root phenotypes in low N + P

The importance of colocalizing root foraging and nutrient availability becomes evident when multiple nutrients are limited. Optimal phenotypes in low N + P had integrated strategies optimized for uptake of both N and P. The phenotypes were found to have roots with more node/whorl occupancy and a wide range of growth angles, as were seen in optimal phenotypes in low N. The LRBD was intermediate between those in low P and low N, with shallower roots having more branches than deep roots. This ensured that efficient soil exploitation could occur in regions with greatest availability of P while exploring for nitrate in subsoil. Even though the number of whorls occupied were more in the optimal phenotypes under low N + P, the number of roots per node was low. The different states of the number, growth angle, and LRBD of roots at different whorls in bean ensured that the root system had a wide range of angles for exploring maximum soil volume. Another strategy was to have roots that were neither deep nor shallow but intermediate angled roots. BR in bean and NR in maize with an optimal number of branching were found to assume intermediate angles when both N and P were limiting. Dimorphic root architectures with axial roots with greater range of growth angles, or comprising of specific combinations of topsoil foraging such as HBR with traits for subsoil foraging such as steep axial growth angles are thought to be efficient for uptake of P and N (Miguel et al., 2013; Miller et al., 2003). Maize roots with early shallow and late deep rooting are dimorphic (Lynch, 2019). Dimorphic root systems (Burridge et al., 2020; Lynch, 2019) with shallow and deep roots are also efficient for uptake of mineralized and leached N (Ho et al., 2005). A phenotype with greater LRBD in the topsoil and fewer LRBD in the subsoil along the same axial root is thought to be important (Kong et al., 2014); however, such phenotypes were not seen in our simulations because plasticity was not included in our simulations. The utility of plasticity varies depending on the environment and is poorly understood (Schneider & Lynch, 2020). The phene states occupied by the various phenes were intermediate in terms of LRBD as well as angles, more whorl/node occupancy, and fewer roots resulted in many combinations of phenes resulting in a greater number of optimal phenotypes than in low P or low N.

Phenotypes with large diameter/low carbon cost

Optimal phenotypes had a small root diameter with those under low N having a slightly larger diameter than those in low P. While small diameter roots are cheaper to construct and maintain, large diameter roots may have better penetrability and are useful under drought stress (Klein et al., 2020) and are better for mycorrhizal colonization (Reinhardt & Miller, 1990). Development of a strong, large diameter PR, imposed carbon constraints such that only roots with low LRBD could have enough growth to explore and exploit efficiently the nutrient resources needed to accumulate optimal biomass. Investing in axial roots rather than high LRBD thereby reduced the carbon requirement of the total root system, while at the same time enabling much better development of the PR and more seminal and/or nodal axes tending towards greater soil exploration. Root diameter is an aggregate trait by itself, comprised of several anatomical phenes that can further be optimized to reduce the carbon cost, including root cortical aerenchyma, living cortical area, cortical cell file number, and cortical cell size many of which have trade‐offs between nutrient and water acquisition, mechanical strength of root structure, and susceptibility to microbial colonization (Galindo‐Castañeda et al., 2018; Lynch, 2013, 2019). In dicots such as bean that undergo radial growth, large diameter phenotypes can benefit by phenes such as root etiolation (reduced secondary growth), which reduces root metabolic costs (Strock et al., 2018).

Phenotypes with greater root length at depth

Our optimization included maximizing root length deeper than 1 m as one of the objectives. The phenotypes were evaluated after 40 days of growth. At 40 days all optimal phenotypes except those with large diameter PR had roots deeper than 1 m; however, only phenotypes that had greater PR or SR LRBD had more root length beyond 1 m. NR of phenotypes with low LRBD of NR and steep growth angles were found at depths >70 cm, suggesting these roots could contribute to deep soil exploration over time. Early fast root proliferation could improve nutrient capture and vigorous growth during early development helps better establishment of the plant. Phenotypes with deeper roots are better for capturing deeper soil resources (Lynch, 2019, 2021; Lynch & Wojciechowski, 2015). Roots contribute to soil organic carbon in the form of exudates, mucilage, and as fine root turnover decreases with soil depth, deeper roots can contribute to carbon sequestration as well (Kell, 2011; Kell, 2012; Pierret et al., 2016).

In this study, we focused only on a small subset of data corresponding to regions low in N, P, and low N + P, as N and P limitations are ubiquitous in natural soils, are primary constraints to food production in low‐input agroecosystems, and are primary causes of environmental pollution in high‐input agroecosystems (Lynch, 2019). However, the ultimate landscape of all possible constraints faced by a plant in an environment is highly complex and multidimensional. Optimal phenotypes will be different for different soil types/precipitation scenarios, as the utility of root traits are dependent on the pattern of water availability in the target environment (Dathe et al., 2016), seasonal rainfall distribution, soil type, crop management, etc. (e.g., Lilley & Kirkegaard, 2011; Lynch et al., 2021) and depend on biotic factors, root loss as well as competition among plants of the same species as well as other species. Many chemical and physical constraints occur in the subsoil, which effectively reduce rooting depth, water use, and nutrient acquisition. Occurrence of several constraints simultaneously requires the integration of several distinct phene states in one optimal phenotype specific to the target environment. The use of FSPM with MOEA provides a valuable tool to identify phenotypes specific to target environments.

Future directions

Understanding the root phenome is a bottleneck to breeding crops with improved nutrient efficiency and stress tolerance. The complexity of fitness landscapes and inability of plant biologists and crop breeders to explore the phenotypic space through empirical experimentation is a major constraint to the design of breeding strategies for complex phenotypes. The focus on identifying useful phenotypes has been limited to evaluating a specific phene state or small set of phene states rather than a large number of integrated phenotypes. Our approach of combining a mechanistic model of root architecture with an evolutionary algorithm can be useful in providing information for selecting and breeding for a limited number of distinct root phenotypes. These results identify phenotypes that have specific elements of ideotypes confirmed to have utility for improved P acquisition or N capture. Many optimal phenotypes identified by the optimization algorithm are phenotypes integrating specific nutrient acquisition strategies previously identified empirically. The algorithm results in several alternate phenotypes across the NP landscape, all of which have not been included in this study. A wealth of information is made available by the MOEA, which can be further used to study integrated phenotypes across different regions of the NP landscape as well get insights into the mechanisms of phene interactions. Including several other parameters of agronomical interest can expand the utility of the framework to identify optimal phenotypes across various constraints.

EXPERIMENTAL PROCEDURES

Description of the model

The functional–structural plant model SimRoot, which has been used successfully to simulate the root growth of several crop species, including bean, maize, barley, and squash (Lynch et al., 1997; Postma et al., 2017; Postma & Lynch, 2012; Rangarajan et al., 2018) was used in this study. SimRoot is now an open‐source platform, OpenSimRoot (Postma et al., 2017). SimRoot (Lynch et al., 1997; Postma & Lynch, 2011a; Rangarajan et al., 2018) was used in conjunction with the Borg MOEA (Hadka & Reed, 2013) algorithm.

SimRoot simulates root architecture in three dimensions in vector space and represents the root system by connected root nodes. Connected root nodes form roots of specific root classes, which form the whole root system. Properties such as nutrient uptake are calculated for each root node and integrated over the length of the root system. SimRoot reconstructs root system architecture from empirical data such as growth rates, angles, and branching frequencies of different root classes. The soil domain is simulated by a finite element model where each node contains values for water and nutrient content. Mass flow and diffusion of P in the rhizosphere around the root is simulated by Barber–Cushman's model (Itoh & Barber, 1983). As the Barber–Cushman model is a one‐dimensional radial model, to account for inter‐root competition, the average mid‐distance between the roots in the vicinity of each root segment was used as the boundary across which nutrient flux is assumed to be zero. As new roots grow in the neighborhood of existing roots, the mid‐distance is adjusted. The initial concentration of nutrients that is available for the new root is corrected for nutrient extraction by existing roots. Water flow is simulated using the Richard's equation and nitrate movement (using the convection dispersion equation) in the soil domain is simulated using SWMS_3D (Šimůnek et al., 1995). Uptake of nitrate in the root is based on Michealis–Menten kinetics similar to that in the Barber–Cushman model. The nitrate concentration at the root surface is a distance‐weighted average of the nitrate concentration at the neighboring finite element nodes and uptake by the root nodes are distributed over the finite element nodes accordingly. A cubic finite element grid with a resolution of 1 cm was used as this resolution was found to provide the best balance between speed and numerical accuracy (Postma et al., 2014). A detailed description of the various processes in SimRoot can be found in Postma and Lynch (Postma & Lynch, 2011a, 2011b; Postma et al., 2014; Dathe et al., 2016; Rangarajan et al., 2018).

All the simulations were conducted in a single soil environment corresponding to a wet silt loam soil. The different environments simulated varied only in the availability of nutrients, nitrate and P. P was vertically stratified with the greatest P availability in the top 10 cm of the soil. Nitrate is initially in the topsoil but leaches to the deeper strata over time with precipitation events. A single precipitation regime was used in all simulations. The initial relative distribution of the nutrients in the soil was kept the same and availability of nutrient was varied by varying initial nutrient availability. Several parameters in SimRoot are not single values but distributions and this causes some stochasticity in the root system and model outputs (Postma et al., 2014). For this study, we removed stochasticity in the parameters as the optimization routine determines optimal phenotypes in subsequent generations based on SimRoot outputs.

The Borg MOEA is a many objective, multimodal optimization procedure (Hadka & Reed, 2013). It represents a class of algorithms whose operators are auto‐adapted based on the problem search progress feedback combines ɛ‐dominance archiving, ɛ‐progress, and randomized restarts (Hadka & Reed, 2013). The algorithm includes an ɛ‐box dominance archive for maintaining convergence and diversity throughout the search, use of ɛ‐progress, which is a computationally efficient measure of search progression and stagnation, an adaptive population sizing operator to maintain search diversity and to facilitate escape from local optima, and multiple recombination operators to enhance the search in a wide assortment of problems.

Figure S8 outlines the functioning of the SimRoot‐Borg simulation–optimization system. The parameters explored (also called input variables or decision variables) are root phene states, including angles, number of roots, and LRBD. The numerical outputs from the SimRoot model are used as the objectives subjected to optimization. The constraints on the range of values a decision variable can assume is set based on studies on root trait variations derived from phenotypic studies in published literature and this defines the space to be explored within a given domain of variation. The input variables for the maize root system and bean root phenotypes included in the study with the constraints on the range of values are presented in Tables S1 and S2 respectively.

Population size and the number of generations were chosen after performing many simulations and taking into account the needs of our case study. Preliminary studies showed that the optimization runs headed towards regions of high nutrient availability. To include all regions of the nitrate P availability landscape, availability was included as an objective that was minimized. Optimization runs were conducted on the Texas Advanced Computing Center's Stampede and Cornell University's The Cube. In total, 50 000 runs corresponding to 500 generations were run with at least five random seed resulting in at least 250 000 total evaluations each for the bean system and maize system and the solutions from the end of the run with each seed were used for further analysis. Epsilon values corresponding to 10% of objective values were used. Solutions from specific regions in the nitrate and P landscape were selected for further analysis. The regions included correspond to regions with low P and non‐limiting nitrate, low nitrate and non‐limiting P, and regions where both nitrate and P were limiting. The algorithm was parameterized according to the recommendations in Hadka and Reed (2015).

Analysis of simulation results

Model outputs and visualization of the objective space

Self‐organizing map was performed to analyze the objective space within the Pareto‐optimal set of solutions. The Pareto‐optimal solutions consist of a variety of distinct phenotypes that differ in their performances in one or more objectives. Self‐organizing maps provide a graphical and qualitative way of extracting knowledge. SOMs result from a process in which neighboring clusters influence each other, resulting in a network topology reminiscent of biological systems (Kohonen, 1998; Wehrens & Buydens, 2007). A SOM allows the projection of information embedded in the multidimensional objective and decision spaces on to a two‐dimensional map (Bandaru et al., 2017). All phenotypes, regardless of the region in the nitrate‐P landscape they evolved in, were clustered under a SOM scheme (som function) and 0.1000 training iterations were used during clustering, over which the α‐learning rate decreased from 0.05 to 0.01. Phenotypes are thus assigned to a node in the SOM grid based on their combined performance in all the objectives. In this way different phenotypes in the Pareto front are clustered solely based on their position in the objective space. Phenotypes evolved in different regions of the nitrate/P landscape having similar performances in all the objectives were clustered on the same or neighboring nodes by this method.

Analysis of optimal phenotypes

Nodes containing phenotypes with greater shoot biomass under each combination of available nitrate and P was considered for further analysis. The optimal phenotypes resulting from the optimization procedure are obtained as vectors of numerical values of root traits corresponding to each root type. These values are in a continuous space and to represent them, a heatmap plot was used. The root systems were simulated based on the optimized root phene values and images rendered for visualizing the root phenotype.

Several phenotypes were seen in the optimal set. Three phenotypes were selected for further analysis in each region of the NP landscape (low P, low N, low N + P). A sensitivity analysis was conducted on the phenotypes evolved in the low N region by varying the number of NR.

AUTHOR CONTRIBUTIONS

HR conducted the experiments, analyzed the data and wrote the article. DH conducted the experiments and analyzed the data. PR supervised the project and contributed to data analysis and writing. JL conceived and supervised the project and contributed to data analysis and writing. JL agrees to serve as the author responsible for contact and ensures communication.

CONFLICT OF INTEREST

The authors declare that they have no competing interests.

Supporting information

Figure S1. Distribution of each objective in bean SOM map (a). Distribution of each objective in maize SOM map (b).

Figure S2. Mean value of objective in each node for bean optimal phenotypes in a region with suboptimal N and P (a). The relative performance of the phenotypes in different objective in each node (b). Nodes 6, 8, and 9 have comparable biomass but vary in performance in other objectives. Some representative bean phenotypes with comparable biomass from nodes 6, 8, and 9 (c).

Figure S3. Mean value of objective in each node for maize optimal phenotypes in a region with suboptimal N and P (a). The relative performance of the phenotypes in different objective in each node (b). Nodes 1, 2, 3, 5, and 6 have comparable biomass but vary in performance in other objectives. Some representative phenotypes with comparable biomass from nodes 1, 2, 3, 5, and 6 (c).

Figure S4. Different primary root phenotypes found in optimal bean phenotypes (a) and optimal maize phenotypes (b).

Figure S5. Different hypocotyl‐borne root phenotypes found in optimal bean phenotypes.

Figure S6. Different basal root phenotypes found in optimal bean phenotypes.

Figure S7. Different nodal root phenotypes found in optimal maize phenotypes.

Figure S8. Flow chart of the SimRoot‐Borg loop.

Table S1. Range of input values for generating bean root phenotypes.

Table S2. Range of input values for generating maize root phenotypes.

Click here for additional data file. (3MB, pdf)

ACKNOWLEDGMENTS

This research was supported by the Foundation for Food and Agriculture Research ‘Crops in Silico’ project, National Institute of Food and Agriculture, U.S Department of Agriculture Hatch project 4732 and USDOE ARPA‐E Award Number DE‐AR0000821.

DATA AVAILABILITY STATEMENT

The data corresponding to the optimal phenotypes used and presented in the analysis has been hosted in https://figshare.com/s/aeb3812cfcd2de6ec4f4. The Borg‐MOEA code cannot be shared openly due to patent protections but code is made freely available for non‐profit research and educational uses and can be obtained upon request (http://borgmoea.org/).

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Figure S1. Distribution of each objective in bean SOM map (a). Distribution of each objective in maize SOM map (b).

Figure S2. Mean value of objective in each node for bean optimal phenotypes in a region with suboptimal N and P (a). The relative performance of the phenotypes in different objective in each node (b). Nodes 6, 8, and 9 have comparable biomass but vary in performance in other objectives. Some representative bean phenotypes with comparable biomass from nodes 6, 8, and 9 (c).

Figure S3. Mean value of objective in each node for maize optimal phenotypes in a region with suboptimal N and P (a). The relative performance of the phenotypes in different objective in each node (b). Nodes 1, 2, 3, 5, and 6 have comparable biomass but vary in performance in other objectives. Some representative phenotypes with comparable biomass from nodes 1, 2, 3, 5, and 6 (c).

Figure S4. Different primary root phenotypes found in optimal bean phenotypes (a) and optimal maize phenotypes (b).

Figure S5. Different hypocotyl‐borne root phenotypes found in optimal bean phenotypes.

Figure S6. Different basal root phenotypes found in optimal bean phenotypes.

Figure S7. Different nodal root phenotypes found in optimal maize phenotypes.

Figure S8. Flow chart of the SimRoot‐Borg loop.

Table S1. Range of input values for generating bean root phenotypes.

Table S2. Range of input values for generating maize root phenotypes.

Click here for additional data file. (3MB, pdf)

Data Availability Statement

The data corresponding to the optimal phenotypes used and presented in the analysis has been hosted in https://figshare.com/s/aeb3812cfcd2de6ec4f4. The Borg‐MOEA code cannot be shared openly due to patent protections but code is made freely available for non‐profit research and educational uses and can be obtained upon request (http://borgmoea.org/).

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