A Mathematical Framework for Analyzing Wild Tomato Root Architecture

Abstract

The root architecture of wild tomato, Solanum pimpinellifolium, can be viewed as a network connecting the main root to various lateral roots. Several constraints have been proposed on the structure of such biological networks, including minimizing the total amount of wire necessary for constructing the root architecture (wiring cost), and minimizing the distances (and by extension, resource transport time) between the base of the main root and the lateral roots (conduction delay). For a given set of lateral root tip locations, these two objectives compete with each other—optimizing one results in poorer performance on the other—raising the question how well S. pimpinellifolium root architectures balance this network design trade-off in a distributed manner. In this study, we describe how well S. pimpinellifolium roots resolve this trade-off using the theory of Pareto optimality. We describe a mathematical model for characterizing the network structure and design trade-offs governing the structure of S. pimpinellifolium root architecture. We demonstrate that S. pimpinellifolium arbors construct architectures that are more optimal than would be expected by chance. Finally, we use this framework to quantify structural differences between arbors grown in the presence of salt stress, classify arbors into four distinct architectural ideotypes, and test for heritability of variation in root architecture structure.

1. Introduction

Root architecture in plants is driven by the development of lateral roots that branch off of the main root (Nibau et al., 2008). Our goal is to derive a model that explains the principles governing branching patterns in Solanum pimpinellifolium root architecture. Root architecture design that affords efficient resource transport is crucial to the overall productivity of the plant (Lynch, 1995); however, it can be advantageous to minimize the amount of material required to construct arbors due to environmental constraints such as resource limitations (Niu et al., 2013), soil compaction (Chen et al., 2014), growth environment orientation (Chiatante et al., 2003), and salt stress (Julkowska et al., 2017). Prior study models plant arbors as a weighted graph, and uses the theory of Pareto optimality to resolve the trade-off between minimizing resource transport delay and minimizing material cost (Conn et al., 2017a,b, 2019). We extend prior study by applying a model developed for Arabidopsis to quantify design trade-offs in S. pimpinellifolium.

Root architecture is highly plastic, showing adaptability to the environments and stress conditions such as salt stress (Julkowska et al., 2014). These salt-induced changes in root architecture were previously used to identify genetic components contributing to overall stress-resilience (Julkowska et al., 2017). Through screening S. pimpinellifolium, which is stress-resilient relative to cultivated tomato, we aim to identify novel strategies that contribute to S. pimpinellifolium resilience. Through modeling and analyzing network topology, we attempt to gain further insight into network efficiency and adaptability to stress conditions.

In addition to observing high stress-resilience, we find that S. pimpinellifolium arbors can be visually classified into four distinct architectural ideotypes: “Christmas Tree,” “Telephone Pole,” “Droopy Telephone Pole,” and “Broomstick” (Fig. 1A). However, a preliminary analysis revealed that this classification scheme is not associated with root structure variation according to traditional physical features (Fig. 1B–D). Our goal was to study alternate methods for quantifying network topology, to test whether our qualitative classification scheme captures meaningful variation in root architecture.

FIG. 1.

Solanum pimpinellifolium root ideotypes across the diversity panel. (A) The four ideotypes of root system architecture were identified in representative accessions: M150/LA1686 for Christmas Tree; M153/LA1689 for telephone pole; M125/LA1628 for broomstick; M298/LA3160 for droopy telephone pole. The pictures represent the 12 days old seedling root system architecture grown under nonstress conditions. We compared S. pimpinellifolium ideotypes for (B) number of lateral roots, (C) average lateral root length, and (D) length of the branched zone. The individual ideotypes are Christmas tree (C), T, dT, and B (B). The asterisks indicate significant differences between pairs of ideotypes using Tukey's honest significance test. In all cases, at least one pair of ideotypes is not detected as significantly different from each other. B, Broomstick; dT, Droopy Telephone Pole; LR; MR; ns; T, Telephone Pole.

Our article is organized as follows. First, we define a graph-theoretic model S. pimpinellifolium arbors that formalizes the objectives of wiring cost and conduction delay. We demonstrate how to resolve the trade-off between these two objectives using the theory of Pareto optimality. We use this framework to investigate how well S. pimpinellifolium arbors optimize wiring cost and conduction delay, and infer whether these objectives are significant drivers of root architecture topology. We then investigate variation in how arbors prioritize objectives. We test whether arbors prioritize objectives in the presence of a salt stressor; we propose a scheme for classifying arbors into four architectural ideotypes, and use network design trade-offs to validate this classification scheme. Finally, we explore whether genetic variation is associated with arbors prioritizing objectives differently.

1.1. Related study

In modeling and analyzing the network topology of S. pimpinellifolium, we relate our study to the Euclidean Steiner tree optimization problem (Agrawal et al., 1995; Arora, 1996; Karpinski and Zelikovsky, 1997). We study the Steiner tree problem with certain constraints on the solution space; in addition, our Steiner trees must balance two cost functions instead of one.

Our study is also closely related to the problem of finding a Light Approximate Shortest-Path Tree (Khuller et al., 1995; Nguyen and Xu, 2007). Given a rooted graph, the goal is to find a spanning tree whose total weight is not too high while maintaining short paths from the root to each vertex. Our study attempts to solve a similar problem but allows for branch points that are not part of the original network.

2. Methods

Hereunder we describe how we use graph theory, as well as the principle of Pareto optimality, to model S. pimpinellifolium root architecture and quantify design trade-offs.

2.1. A graph-theoretic framework for studying S. pimpinellifolium architecture

We model the S. pimpinellifolium root architecture as a weighted connected acyclic graph $A = (V, E)$ . The main root must be contiguous, and every lateral root must have a path to the main root base. This means that there will be a path between any two vertices, because any lateral root can reach any other one by going through the main root. The graph must be acyclic because we do not observe self-loops in root architecture.

Each of the vertices $v \in V$ represents the three-dimensional coordinate of one of the points on the root. We partition the vertices into two disjoint sets: The main root vertices $R \subset V$ , and the lateral root vertices $L = V ∖ R$ .

The main root vertices from a path graph—that is, they can be ordered $R = {r_{1}, r_{2}, \dots, r_{n}}$ such that $(r_{i}, r_{i + 1}) \in E$ for all $1 \leq i < n$ . We designate r₁ as the main root base.

Each lateral root vertex has degree 1, and is directly connected to one of the main root vertices. Lateral root vertices do not connect to another lateral root segment.

For each edge $(u, v) \in E$ , let the weight $w (u, v)$ be the Euclidean distance between u and v. For each pair of points $u, v \in V$ , let the distance $d (u, v)$ be the length of the (unique) path from u to v.

Let the wiring cost be the sum of the edge lengths of all edges in the graph:

Let the conduction delay be the sum of graph distances from the main root base to each lateral root:

2.2. Pareto-optimal root architecture problem

In general, it is impossible to design a root architecture that simultaneously minimizes wiring cost and conduction delay. Furthermore, the trade-off between these two objectives can become arbitrarily large (Fig. 2). To resolve this trade-off, we turn to the principle of Pareto optimality (Da Cunha and Polak, 1967) to define the problem of constructing an optimal arbor.

FIG. 2.

Trade-off between wiring cost and conduction delay. In this example, there are two lateral roots that seek to connect to the main root. The main root base is $1 0^{n}$ units long. Each lateral root is $δ \approx 0$ units from the main root edge and approximately $1 0^{n}$ units from the main root base. The minimum wiring cost arbor (orange) has total cost $1 0^{n} + 2 δ$ ; the minimum conduction delay arbor (purple) has total cost $3 \cdot 1 0^{n}$ . One can generalize this pattern to have k lateral nodes and a wiring cost of $1 0^{n}$ + $k δ$ and a delay of $k \cdot 1 0^{n}$ .

As input, we are given the following (Fig. 3A):

FIG. 3.

Pareto front of optimal solutions. (A) Initially we are given a set of line segments comprising the main root, as well as several lateral root tips that must be connected to the main root. Three possible optimal architectures are shown: one that minimizes the total material cost, one that minimizes the time to transport resources to the main root base, and one that attempts to strike a balance between the two objectives. (B) The Pareto front of optimal solutions. For a given set of points, we vary $α \in [0, 1]$ to compute the set of architectures that achieve an optimal trade-off between the two objectives. The red “X” denotes the costs of an observed arbor; we can measure how close the arbor was to being Pareto-optimal, and the Pareto-optimal tree to which the observed arbor was most similar.

A set of main root points ${r_{1}, r_{2}, \dots, r_{n}}$ that are all connected to each other; in other words, our network starts with the edges ${(r_{1}, r_{2}), (r_{2}, r_{3}), \dots, (r_{n - 1}, r_{n})}$

A set of lateral root tips ${l_{1}, l_{2}, \dots, l_{m}}$ . Initially these points are not connected to the main root.

A value $α \in [0, 1]$ .

We then construct an architecture graph A by connecting each lateral root to the main root (Fig. 3A). Each lateral root may connect to one of the initial main root points, or it may connect to a midpoint on one of the edges $(r_{i}, r_{i + 1})$ (thus splitting that edge into two smaller edges). The architecture graph A should minimize $α \cdot D (A) + (1 - α) \cdot W (A) .$ (3)

The parameter $α$ determines how much the solution should prioritize one objective over the other. Different choices of $α$ will yield different optimal solutions $A_{α}$ that the arbor could have constructed (Fig. 3A). We call these optimal architectures the Pareto front of optimal solutions.

2.3. An algorithm for generating optimal root architectures

Because lateral roots cannot connect to each other, we may simply optimize each lateral root independently. Because the main root is fixed, and because lateral roots have degree 1, optimizing a lateral root reduces to finding the best straight line between that lateral root and a point along one of the main root line segments. This can be done analytically using calculus and linear algebra.

For this section, we will put variables in $b o l d$ to indicate vector-valued variables, and we will use $\cdot$ to represent vector dot products.

A line segment is defined by its two endpoints $P_{0}, P_{1}$ . The slope of the line segment can be represented as a vector $M = P_{1} - P_{0}$ . Any point P that is on the line segment can be represented as $P_{0} + M \cdot t$ for some $t \in [0, 1]$ . We will use this notation to represent each line segment.

Consider a lateral root located at the point $Q$ , and consider a main root line segment $(P_{0}, P_{1})$ with slope $M = P_{1} - P_{0}$ . We will derive a formula for the optimal point to connect $Q$ along this line segment. Initially, consider the case where $P_{0}$ is located at the origin. Then every point on the line segment can be represented as $P_{0} + M \cdot t = M \cdot t$ for some $t \in [0, 1]$ .

If we connect $Q$ to a point $M \cdot t$ along the line segment, the increase wiring cost is $α$ times the distance from $Q$ to $M \cdot t$ . This is $W = α \sqrt{(Q - M t) \cdot (Q - M t)} .$

The increase in conduction delay is $(1 - α)$ times the distance from $Q$ to $M \cdot t$ , as well as the distance from $M \cdot t$ to the origin. This is $D = (1 - α) \sqrt{(Q - M t) \cdot (Q - M t)} + (1 - α) \sqrt{M t \cdot M t} .$

= (1 - α) \sqrt{(Q - M t) \cdot (Q - M t)} + (1 - α) \sqrt{t^{2} \cdot M \cdot M} .

= (1 - α) \sqrt{(Q - M t) \cdot (Q - M t)} + (1 - α) t \sqrt{M \cdot M} .

The total increase in cost is $C = W + D .$

= \sqrt{(Q - M t) \cdot (Q - M t)} + (1 - α) t \sqrt{M \cdot M} .

We then seek to find the value of $α$ which minimizes C. We do so by solving for the value of t, which makes the derivative of C equal to 0: $\frac{d C}{d t} = \frac{1}{2} \times \frac{1}{\sqrt{(Q - M t) \cdot (Q - M t)}} \times (- 2 M \cdot (Q - M t)) + (1 - α) \sqrt{M \cdot M} .$ $= - \frac{M \cdot (Q - M t)}{\sqrt{(Q - M t) \cdot (Q - M t)}} + (1 - α) \sqrt{M \cdot M} = 0 .$

We then rearrange the equation to isolate t:

To make the equation easier to read, we define the following variables:

$x = M \cdot M .$

$y = Q \cdot M .$

$z = Q \cdot Q .$

This gives us $\frac{y^{2} - 2 x y t + x^{2} t^{2}}{z - 2 y t + x t^{2}} = γ x .$

y^{2} - 2 x y t + x^{2} t^{2} = γ x z - 2 γ x y t + γ x^{2} t^{2} .

(x^{2} - x^{2} γ) t^{2} + (2 γ x y - 2 x y) t + (y^{2} - γ x z) = 0 .

Once again, we will define new variable substitutions to make the equation easier to read: $a = x^{2} - x^{2} γ .$ $b = 2 γ x y - 2 x y .$

c = y^{2} - γ x z .

Then we have $a t^{2} + b t + c = 0 .$

To solve for t, we simply use the quadratic formula: $t = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- (2 γ x y - 2 x y) \pm \sqrt{{(2 γ x y - 2 x y)}^{2} - 4 (x^{2} - x^{2} γ) (y^{2} - γ x z)}}{2 (x^{2} - x^{2} γ)} .$

Note that this formula is only valid for $t \in [0, 1]$ . Thus, this formula produces up to two critical points. We test these critical points, as well as the boundary points $t = 0$ and $t = 1$ , to find the value $\hat{t}$ that produces the minimum cost. We then take $M \cdot \hat{t}$ as our optimal midpoint.

Up to this point, we have assumed that our line segment $(P_{0}, P_{1})$ starts at the origin (i.e., $P_{0} = 0$ ) to simplify the algebra. However, we can easily translate all three points, so that $P_{0}$ is at the origin. We can compute translated points ${P_{0}}^{'}, {P_{1}}^{'}, Q^{'}$ such that ${P_{0}}^{'}$ is at the origin and the other two points have been translated the same amount in each direction. Note that because $P_{0}$ and $P_{1}$ have been translated the same amount in each direction, our slope vector $M = P_{1} - P_{0} = {P_{1}}^{'} - {P_{0}}^{'}$ is unchanged. We compute $\hat{t}$ for ${P_{0}}^{'}, {P_{1}}^{'}, Q^{'}$ , and then plug this back in with our original points to get the optimal midpoint of $P_{0} + M t .$

Finally, consider a lateral root at point $Q$ , and a set of main root segments . We compute the optimal midpoint for each line segment $(P_{i}, P_{i + 1})$ ; finally, we connect $Q$ to the best midpoint among all of the possible midpoints that were computed.

We repeat this process independently for each lateral root. In this way, we may analytically compute the optimal arbor for a given value of $α$ .

2.4. Quantifying network design trade-offs in observed architectures

Given an observed architecture A, we may use the aforementioned algorithm to compute the set of optimal root architectures: ${A_{α} | α \in [0, 1]}$ (Fig. 3A, B). Using this, we can calculate two metrics for quantifying network design trade-offs in the original observed arbor (Fig. 3B):

Pareto front distance: The L-2 distance between costs of arbor A and the costs of the closest optimal tree:

{min}_{α} ∥ (D (A), W (A)) - (D (A_{α}), W (A_{α})) ∥ .

This measures how close the original arbor came to managing an optimal trade-off between wiring cost and conduction delay.

Pareto front location: The value of $α \in [0, 1]$ associated with the optimal arbor closest in cost to the observed arbor.

The closer this value to 1, the more strongly the arbor prioritized minimizing wiring cost at the expense of conduction delay.

Under this framework, we tested four hypotheses: (1) S. pimpinellifolium roots are closer to being Pareto optimal than would be expected by chance; (2) S. pimpinellifolium roots respond to salt stress by modifying network design trade-offs; (3) variation in ideotype classification is associated with variation in Pareto front location; (4) Pareto front location is a heritable trait that may be driven (in part) by genetic factors.

2.5. Comparing observed arbors against a null model

Using the aforementioned mathematical framework, we sought to confirm whether S. pimpinellifolium root architecture structure is dictated in part by the principles of wiring cost and conduction delay (as opposed to some other optimization criteria). We did so by comparing S. pimpinellifolium root architecture with the arbors that were Pareto optimal for these objectives. If wiring cost and conduction delay are drivers of S. pimpinellifolium root architecture, then observed arbors should be more similar to the optimal arbors than would be expected by chance.

To evaluate how well S. pimpinellifolium would be expected to optimize wiring cost and conduction delay by chance, we also compared observed arbors with arbors that were randomly generated on the same set of points. This allowed us to establish a baseline for how similar S. pimpinellifolium arbors would be to optimal arbors even if they were explicitly not optimizing wiring cost or conduction delay. To generate a random arbor, we start with the same main root segments, and same lateral root tip locations, as the original observed arbor. Each lateral root tip is then connected to a point chosen at uniform random from the set of line segments comprising the main root.

3. Empirical Data

The natural diversity panel 260 accessions of S. pimpinellifolium was screened for salt-stress-induced changes in root system architecture. The seeds were surface sterilized using 50% bleach for 10 minutes and washed 4–6 times using sterile MiliQ water. The seeds were stratified at 4°C for 24 hours to improve germination. The seeds germinated on square petri dishes (12 × 12 cm) on 1/4 Murashi–Skoog media (1.1 g MS, 5 g sucrose, 1 g M.E.S. buffer, pH 5.8 KOH, and 10 g Dashin agar). Four days after germination, the seedlings were transferred to square petri dishes containing 1/4 Murashi–Skoog media for Control treatment or 1/4 MS media supplemented with 100 mM NaCl for Salt treatment. The plates were scanned every day for 7 consecutive days.

The root system architecture was quantified from the images recorded 0, 1, 2, 3, and 4 days after transfer to Control and Salt plates using SmartRoot software (Lobet et al., 2011). The points (x–y coordinated) representing the observed root system architecture are obtained using SmartRoot. The nodes are added by the software automatically every time a slight change in the root trajectory is observed (Fig. 4). Addition of the nodes in between the observed nodes on the main root trajectory would not affect either the overall shape or size of the root architecture or the number of lateral roots for our model. The global root data were compiled and analyzed in R for lengths of main and lateral roots as well as growth rates of main root and lateral roots. The root node data were used for calculating the Pareto optimality.

FIG. 4.

Graphic representation of the collected root system architecture coordinates. The position of the nodes corresponding to the locations of the main root (yellow) and lateral roots (green) were collected by using SmartRoot software. The position of each node is added whenever the slight change in the root trajectory is observed, to ensure the most precise tracking.

Plants were manually and qualitatively classified into one of four representative ideotype accessions (Fig. 1A). The root systems where lateral roots were decreasing in length throughout the main root axis were classified as “Christmas Tree.” The root systems where the lateral root was constant throughout the main root axis and where lateral roots were growing at ∼90° angle from the main root axis were classified as “Telephone Pole.” The root systems similar to “Telephone Pole,” but with lateral roots growing under ∼45° angle from the main root axis were classified as “Droopy Telephone Pole.” The root systems where majority of the lateral roots originated at the root base were classified into “Broomstick” ideotype.

4. Results

4.1. S. pimpinellifolium roots are Pareto optimal

For each arbor, we computed the Pareto front of optimal architectures that the arbor could have constructed, and measured each arbor's distance to its respective Pareto front. To test whether arbors are closer to being Pareto optimal than would be expected by chance, we compared each arbor against 20 randomly generated arbors (Fig. 5A). We find that S. pimpinellifolium arbors are more optimal than random trees 98.61% of the time, which is significantly >50% (binomial test, $p < 1 0^{- 322}$ ). On average, the S. pimpinellifolium arbor is $3.15 \pm 2.60$ times closer to the Pareto front than the random arbor; this ratio is significantly >1 (Welch's t-test; $p < 1 0^{- 322}$ ). We conclude that S. pimpinellifolium architecture optimizes wiring cost and conduction delay significantly more than would be expected by chance. This suggests that further study of S. pimpinellifolium may yield new insights into design of distributed network optimization algorithms.

FIG. 5.

Analysis of Pareto front distance and location. (A) For each Solanum pimpinellifolium arbor, we calculated the distance to the Pareto front, as well as the distance for 20 arbors that were randomly generated on the same set of main root and lateral root points. We find that S. pimpinellifolium are significantly closer to being Pareto optimal than would be expected by chance. (B, C) For each arbor, we calculated its closest location on the Pareto front, a measure of how the arbor prioritized objectives. We find significant differences between how arbors grown in different conditions (B), and arbors from different ideotype classes (C), prioritize objectives.

4.2. Salt stressor affects network design trade-offs

Arbors grown in control conditions tended to prioritize optimizing wiring cost more than arbors grown in conditions with a salt stressor (Fig. 5B). The average value of $α$ for arbors grown in control conditions was $0.55 \pm 0.26$ , as compared with $0.50 \pm 0.28$ ; the difference between the two groups was statistically significant (Mann–Whitney test; $p < 0.001$ ). This suggests that salt stressors may lead arbors to place extra emphasis on fast resource transport.

4.3. Classification of ideotypes is associated with variation in network design trade-offs

Finally, we find a strong association between an arbor's architectural ideotype and the degree to which an arbor prioritizes trade-offs between wiring cost and conduction delay (Fig. 5C). There are significant differences in Pareto front location between different ideotype groups (Kruskal–Wallis one-way analysis of variance; $p < 1 0^{- 28}$ ). The Pareto front location was significantly different between all pairs of ideotype groups (Mann–Whitney test with a Bonferroni correction, $p < 0.01$ in all cases). Thus, Pareto front location represents a quantitative measure that is associated with the architectural diversity that we observe upon visual inspection of S. pimpinellifolium arbors.

4.4. Network design trade-offs are heritable

The variation for Pareto front location was lower within a single genotype than the variation observed across individual genotypes. We calculated broad-sense heritability $H^{2} = \frac{V_{G}}{V_{P}}$ , defined as proportion of variation due to genotype specific effects V_G to the observed phenotypic variation across the genotypes V_P; values close to 1 indicate high heritability, and values close to 0 indicate low heritability (Griffiths et al., 2005). Broad-sense heritability for Pareto front location was $0.6345$ for root systems measured 4 days after transfer, indicating the high heritability of this trait and its potential use in forward genetic studies.

5. Discussion

Similar to many natural and engineered systems, S. pimpinellifolium root architecture must balance trade-offs between multiple competing objectives. Generally, there is not a single architecture that is optimal for all possible objectives. Instead, the best the arbor can do is converge to an architecture that is Pareto optimal—meaning it is impossible to improve one objective without sacrificing performance in the other objectives. Our study proposes two objectives—minimizing wiring cost and minimizing conduction delay—that S. pimpinellifolium roots seek to optimize in a Pareto optimal manner. Using graph theory and the principle of Pareto optimality, we define a framework for quantifying the optimality of a given S. pimpinellifolium root architecture.

We apply this model to S. pimpinellifolium imaging data and find that S. pimpinellifolium arbors are significantly closer to being Pareto optimal for these objectives than would be expected by chance. We use Pareto front location to show that S. pimpinellifolium arbors prioritize objectives differently in the presence of a salt stressor. We find that unlike traditional network design metrics, Pareto front location quantifies S. pimpinellifolium architecture in a way that is strongly associated with the architectural diversity we observe qualitatively. Finally, we find that Pareto front location shows a strong association with genetic variation. Overall, our study provides a framework for quantifying S. pimpinellifolium based on how and how well they manage network design trade-offs. More broadly, our study provides a blueprint for quantifying how natural and engineered systems manage trade-offs between several competing objectives.

6. Future Directions

Pareto front location is associated with our classification of S. pimpinellifolium arbors into architectural ideotypes. Future study involves using clustering analysis of Pareto front location to conclusively validate our chosen classification scheme (or identify the correct number of clusters). If we find evidence for the existence of our four identified ideotypes, future study will seek to find the functional basis behind each of these four structures.

In addition to formulating an exact algorithm for generating Pareto-optimal arbors, we seek to study the development process of S. pimpinellifolium. We will take advantage of data tracking the growth of individual arbors over time to measure how arbors prioritization of objectives changes over time, and eventually define a model for the distributed growth process used to generate Pareto optimal architectures.

Future study includes Genome Wide Association Study, to identify underlying genetic components of Pareto front location and distance in S. pimpinellifolium roots. Through reverse genetic approaches we aim to further study genetic components of Pareto front location, their role in root architecture development and further developing better understanding on how S. pimpinellifolium plants prioritize objectives during nonstress and stress conditions.

Finally, we will expand our model to study the trade-off between not only wiring and transport efficiency, but also gravitropism and growth media exploration. Certain arbors may have a higher tendency to grow toward or away from gravitational forces in the growth medium. Furthermore, certain arbors may exhibit higher nutrient uptake at the main root, thus leading lateral roots to explore farther horizontally to avoid competing with the main root. We will expand our mathematical model to include these factors as extra constraints on how lateral roots may connect to the main root (while still optimizing wiring cost and conduction delay.

Footnotes

Authors’ Contributions

Data curation, formal analysis, methodology, software, validation, visualization, and writing (original draft preparation) by A.C. Conceptualization, data curation, formal analysis, investigation, methodology, visualization, and writing (original draft preparation) by M.M.J.

Acknowledgments

We thank the organizing committee for the 2021 Biological Distributed Algorithms (BDA) conference, as well as the reviewers for or BDA submission. We thank Guillaume Lobet for help with understanding the output from the plant imaging software. We thank Graham Zug, who is currently working with Dr. Chandrasekhar on mathematical optimization algorithms for constructing Pareto optimal plant arbors. Finally, we thank bioRxiv for hosting a preprint of our study (DOI: 10.1101/2021.08.12.456185).

Data Availability

We will make data available upon request. Our code for analyzing arbors and performing statistical analysis can be found at .

Author Disclosure Statement

The authors declare they have no competing financial interests.

Funding Information

Dr. Julkowska's study was financed by the King Abdullah University for Science and Technology and the Boyce Thompson Institute.

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