Artificial Plant Root System Growth for Distributed Optimization: Models and Emergent Behaviors

Weixing Su; Lin Na; Fang Liu; Wei Liu; Muhammad Aqeel Ashraf; Hanning Chen

doi:10.1515/biol-2016-0059

Article Open Access

Artificial Plant Root System Growth for Distributed Optimization: Models and Emergent Behaviors

Weixing Su , Lin Na , Fang Liu , Wei Liu , Muhammad Aqeel Ashraf and Hanning Chen

Published/Copyright: December 15, 2016

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From the journal Open Life Sciences Volume 11 Issue 1

Abstract

Plant root foraging exhibits complex behaviors analogous to those of animals, including the adaptability to continuous changes in soil environments. In this work, we adapt the optimality principles in the study of plant root foraging behavior to create one possible bio-inspired optimization framework for solving complex engineering problems. This provides us with novel models of plant root foraging behavior and with new methods for global optimization. This framework is instantiated as a new search paradigm, which combines the root tip growth, branching, random walk, and death. We perform a comprehensive simulation to demonstrate that the proposed model accurately reflects the characteristics of natural plant root systems. In order to be able to climb the noise-filled gradients of nutrients in soil, the foraging behaviors of root systems are social and cooperative, and analogous to animal foraging behaviors.

Keywords: Plant Root Foraging; Root Branching; Distributed optimization; Life-cycle Model; bio-inspired computing

1 Introduction

Recently, a considerable amount of bio-inspired computing studies have been undertaken to exploit the analogy between searching a given problem space for an optimal solution and the natural process of foraging for food [1,2]. These bio-inspired computing approaches are increasingly used by engineers and scientists to solve complex optimization problems that are intractable using conventional methods [3]. Generally, a bio-inspired optimization problem solving process occurs in the following manner: an initial position is randomized in the search space, and its acceptability assessed through the application of a fitness function; a bio-inspired position change strategy, consistent with the paradigm in use, is then iteratively applied with the hope of improving the solution acceptability; the final solution is identified either through achieving an acceptable level of fitness or on the completion of a set amount of computation.

Successful examples of such bio-inspired computing algorithms as evolutionary algorithms [4] include genetic algorithm, evolutionary strategy, evolutionary programming, and genetic programming, ant colony systems [5], particle swarm optimization [6,7], and bee foraging algorithms [8].

Although foraging behavior is a typically considered an animal characteristic, other organisms, including plants, have shown similar traits [9]. Because of plants’ unique non-motile way of life, they only have access to resources nearby their growth site [10]. The above description is the main difference between plant growth and animal foraging. Obviously, the survival rules for each plant species is to efficiently find soil with sufficient nutrients and water. So, the ability of plant roots to sense they myriad factors in their local environment allows them to complete in the evolution process, and the growth direction and root system development are driven by these factors [11].

Continuous changes of the natural environment are considered as the reason of plant root growth diversity, including increased lateral branching, root biomass, root length and uptake capacity. It should be noted that these developmental needs require correct auxin transport and signaling [12]. The number of roots and the length per unit mass of roots also changes in to response to heterogeneity [9]. Many studies have demonstrated that plant foraging is optimal for their local environment, but this assumption is built on scarce experimental evidence. The development of additional optimal plant foraging models requires further research and consideration [13–15].

2 Plant root growth

2.1 Root foraging along nutrition gradient

As the main nutrient supply plants, roots are highly sensitive to the availability of basic resources in the soil. Various types of plant roots can detect several kinds of environmental nutrient gradients, and respond differently by changing their growing orientation to boost the exploration of nutrient-rich areas. Some research has been focused on this growing response of the plant root and referred to it as tropism [16]. Generally speaking, the roots of plants show different characteristics, such as gravitropism, phototropism, hydrotropism, thigmotropism, thermotropism, electrotropism, magnetotropism and chemotropism, in response to environmental gradients of gravity, light, water, touch (mechanical stimuli), temperature, electric field, magnetic field, and chemicals respectively [17–19]. Of these important orientations and motions, geotropism and hydrophilicity are the main factors determining the roots directional growth.

In order to greatly improve soil resource utilization, the plant root system has developed a self-adjusting growth scheme to control main root growth and root bifurcation dynamics. That is, the main roots of different species generally display the same orthogravitopism (i.e., grow downward), whereas lateral roots are always diagravitropism or oblique (i.e., subside extending) [20].

2.2 Root system coevolution

Studies have shown that animal feeding decision is often influenced by both current situation and past experience [21,22]. Previous studies have suggested that plant behavior is simpler than that of animals. Now, however, plant biologists have discovered that experiential accumulation through conditioning can also significantly affect plant behavior [12,23,24]. Plants also have memory and communication behavior, even if they lack a central nervous system. Plants represent memory and behavioral changes depending on their previous experience or their parents’ experience. For example, the growth of the clover branch mainly depends on both its current neighbors and those of the past year [25]. Plants can also communicate with other plants, herbivores, and mutualists [26,27]. For instance, plant competitors exhibit significantly reduced root width and spatial separation in soils with a coherent nutrient distribution; whereas others in soils without uniform distribution moderately reduce their root growth, indicating that the plants integrate information about neighbors and resource distribution in determining root foraging activity.

2.3 Auxin-regulated root tips life-cycle

The main factors for root growth and development is auxin. Accurate auxin transport and signaling can stimulate root development in different stages [28].

The initiation and growth of lateral root is affected by auxin. For example, when Arabidopsis root pericycle cells undergo reprogramming into lateral root, lateral root growth is initiated. At this time, auxin gathers in the pericycle cells adjacent to xylem vessels [29]. Furthermore, auxin has other effects on lateral roots, including the development of root systems, the structure of lateral root primordia, and root branching from parent roots [30]. In addition, because of over-expression of the DFL1/GH3-6 or the IAMT1 genes, which encode enzymes modulating free IAA levels, lateral root formation is reduced [31].

Root branching can rapidly regulate the surface of root systems, and plants have already evolved an effective control mechanism. The lateral root is initiated by auxin transport, a key for root branching regulation of timing and location. During this process, the development of primordia can be arrested for a certain time [32].

3 Root foraging optimizer

In this section, we present a new optimization model based on root-foraging behavior of main plants, called root system growth optimization. This uses root-foraging, memory, communication, and auxin regulatory mechanisms in the root system mentioned in Section 2.

3.1 Auxin model

The primary target here is finding the minimum of F(x), x RD, which indicates the distribution of nutrient in the soil. Here we use F(x) > 0, F(x) = 0, F(x) < 0 to represent three phenomena, the existence of nutrients, a neutral medium, and the existence of noxious substances, respectively.

In the RSGO model, the plant root system is defined as consisting of several root tips. Mathematical expressions can be expressed as follows:

RS={θit|i=1,2,…,Pt;t=1,2,…,T}(1)

Where

θit=〈xit,fit,nit,αit,ϕit〉(2)

Here θit denotes a root tip; P^t denotes the count of the root tips at time t and T denotes the final time of the root system growing process. The root tip θit exhibits five characteristic parameters: space position xit, fitness fit, nutrient nit, the auxin αit, and orientation ϕit, . αit depends upon fit, and nit, which control the tip’s activity, foraging (elongate), branchin, or death at time t. If foraging, the root tip moves with an orientation ϕit as an angle formed by the root axis.

Initially with t=0, a certain number of root tips P0 are initialized randomly in the D-dimensional space. The space position and head tilt angle of the ith root tip is denoted as x_i = (x_i1, x_i2,...x_iD ) and ϕ_i = (ϕ_i1,ϕ_i2,...ϕ_i(D-1)), respectively. Here x_id ∈[l_d , u_d ], d∈[1, D], ld, ud are the lower and upper bounds, respectively. The process of forging represents with mathematical description is that for each time step t, the root tip i will forage for nutrient and its nutrient can be updated by:

nit+1=nit+1iffit+1<fitnit−1else(3)

In the initialization phase, all the root tips of the nutrients are set to zero. During root growth foraging, for each tip in the root system, the root tip is considered to be able to obtain nutrients from the environment and the nutrients are added if the new tip position is better than the last one. Otherwise, the apical foraging process results in the loss of nutrients, its nutrients reduced by one.

Then the auxin concentration αit, which combines the health and energy states of the ith root tip, is manipulated according to the following equations:

healthit=fit−fworsttfbestt−fworstt(4)

energyit=nit−nworsttnbestt−nworstt(5)

αit=ξhealthit∑j=1Pthealthjt+(1−ξ)energyit∑j=1Ptenergyjt,ξ⊂[0,1](6)

where fworstt/fbestt and nworstt/nbestt are the current worst/ best fitness and nutrient of the whole root system at time t.

In each cycle of the root growth process, all root taps are classified according to the auxin concentration values defined in formula (6). That is, a root tap having a high auxin concentration value has a higher probability of being selected as the main root for branching. According to the above idea, the number of the root is calculated as follows:

Smt=Pt×Cr(7)

where Smt denotes the group size of main root being selected, Pt is the entire number of root tips, and Cr is the selection probability. Lateral roots are calculated by Slt=Pt-Smt.

3.2 Root tip branching

The principle of root branching can be summarized as follows: a threshold, BranchG, is defined and compared with the auxin concentration value of each main root to determine whether it performs branching:

nobranchingbranchingotherelseifαit>BranchG(8)

If θit is chosen as a main root and its auxin concentration is adequate to implement branching, the branching number wi of θit is determined by:

Sit=⌈R1αit(Smax−Smin)+Smin⌉(9)

where Smax and Smin denote the maximum and minimum of the new growing tips, and R1 denote a random distribution coefficient. Considering growth direction of θit as reference angle, the searching space of its all branches is divided into Smax subzones and the angle of new growing tips (i.e. the new root branches of θit) is randomly falling within one of these subzones. Then these new branching tips will grow as:

ϕjt+1=ϕit+λjϕmax/Smax(10)

xjt+1=xit+R2lmaxH(ϕjt+1)(11)

where j⊆[Smin,Sit] is the root branch index of root tip θit,λj⊆[1,Smax] is the selecting subzone number of the root branch xjt+1,ϕ_max is the maximum growing turning angle, which is limited to π, R2 is random value between 0 and 1, l_max is the maximum of root elongation length, and Hϕjt+1=hj1t+1,hj2t+1,…,hjDt+1 is a Polar to Cartesian coordinates transformation function, which can be calculated as:

hj1t+1=∏p=1D−1cos⁡(ϕjpt+1)hkt+1j=sin⁡(ϕj(k−1)t+1)∏p=jD−1cos⁡(ϕjpt+1)hjDt+1=sin⁡(ϕj(D−1)t+1)(12)

3.3 Hydrotropism and gravitropism

In section 2, the conclusion that different tropisms can influence root locus is described. The RSGO model can describe hydrotropism and gravitropism. Gravitropism is affected by the communication mechanism in root system. Since half of main roots will grow toward the position with more moisture in the root system, that is given by:

xit+1=xit+R3(xbestt−xit)(13)

where i⊆[1,Smt/2], R3 is a random value in the range (0, 1), and xbestt is the best position in the root tip group.

Considering hydrotropism depends on root memory, as the rest of the main roots will grow along their original directions as:

xit+1=xit+R4lmaxH(ϕit)ifxit>xit−1(14)

where i⊆[Smt/2,Smt],l_max is the maximum of root elongation length, and R4 is random value in the range (0, 1).

3.4 Random walk of lateral roots

The optimal foraging method for random distribution of nutrients at each feeding session is that all lateral apices move randomly [21]. At the t-th iteration, each lateral root tip will randomly generate the head angle and the length of the elongation. The above description is given by:

ϕit+1=ϕit+R5ϕmax(15)

xit+1=xit+R6lmaxH(ϕit+1)(16)

where i⊆[0,Slt], R5 and R6 are random values and the range of values is (0, 1), ϕ_max is the maximum growing turning angle, and l_max is the maximum of root elongation length.

3.5 Root tip death

When apices do not receive more nutrients in their lifetime, they exhibit low auxin concentrations and are therefore not active and are less likely to continue to grow. In addition, when the root growth is at different stages, if the auxin concentration is less than zero, apical death and removal from the root group occurs.

4 Simulation of root foraging

In Section 3, we have obtained an optimization model of the social foraging and plant root growth system. The RSGO model has the ability to simulate long-term root growth and foraging behaviors, which spanning a great deal of generations, and stressing the most important morphological and physiological aspects of a root system. This can be connected by social foraging strategies which have developed over millennia and resulted in distributed non-gradient optimization algorithms which been designed to do global search in the noise space. In this Section, simulations from three different scales are performed and comparfed with real plant species doses. In individual roots, only one tip is initialized as the main root and grows directionally according to gravitropism and hydrotropism feedback and environmental constraints. At the group level, the dynamic property of auxin regulation underlying root tip branching and death are the key-point in the simulation. At last, some of new features about root structure can be obtained through the simulation which focuses on the root system development process in RSGO model.

4.1 Tropism

The most important attribute of plant root evolution is the socialization of their foraging behavior in order to be able to climb the nutrient gradient in soil. The directional growth of plant roots in the direction of environmental stimuli is a kind of tropism. Two important and well-studied tropisms include gravitropism, which determines the growing direction of roots, for example with orthogravitropism or downward growth, and hydrotropism, the ability to apperceive the gradient of environmental humidity for governing the growing direction

Different from some exiting models of plant root growth, the proposed model pays more attention to the social optimal foraging ability, and thus achieves the characteristic of gravitropism and hydrophilic. As discussed in Section 3.3, a simple method to operate gravitropism is to calculate the degree of adaptation of all roots and pick the most appropriate, and this way cause the dominant motion of the root tip group to the moisture area. The implementation of the hydrophilic can result in each root tip can find and climb to the nutrient gradient of the environment.

We have simulated plant root growth behaviors (the hydrotropism and gravitropism of the RSGO model) in a 2D sphere and Griewank environment (as shown in Fig. 1 and Fig. 2). In these two different environments, three independent scenarios are simulated, each with a hydrotropic reaction with gravity-free, gravity reaction with hydrotropism-free, and both reactions We set only one tip as the main root (expressed as the red line in the figure) in each simulation, with the initial position (-3, -3). This main root is also the only branch that can be branched (shown in the figure as a green line) throughout its life cycle.

Figure 1

Simulation on 2-dimensional Sphere considering: (a) hydrotropism; (b) gravitropism; (c) hydrotropism and gravitropism.

Figure 2

Simulation on 2-dimensional Griewang considering: (a) hydrotropism; (b) gravitropism; (c) hydrotropism and gravitropism.

In the 2D sphere and Griewank, the growth trajectories of the hydrotropic reaction in the RSGO model are shown in Fig. 1(a) and Fig. 2(a) respectively. From Fig. 1(a) and Fig. 2(a), we can observe the basic hydrotropic rule in RSGO. The tip rises up the water gradient by adding up branch count and roots extension. It can be clearly seen that the hydrotropic rule in RSGO is a typical local search mechanism, which is well documented by earlier studies in plants and animals: when there are various types of resources with different costs and different benefits, the expected organism chooses the resources which maximize benefits and minimize costs. From the growth trajectories of Fig. 1(b) and Fig 2(b), we can find the root tip shifts over the entire unimodal and multimodal landscape (defined by Sphere and Griewank, respectively) through the gravity in RSGO model. In other words, the gravity designed in RSGO is a global search strategy that makes each tip a social forager to maximize the performance of the root system as a whole, rather than their own personal performance.

Fig. 1(c) and Figure 2(c) justify the root-growing trajectory with the influence of the two elements. In the simulation, we can find that gravitropism is affected by hydrotropism. This gives an explanation of how roots perceive various environmental hints and show different responses to the tropic influences. That is to say, the law of hydrotropic in RSGO encourages mining ability, and the gravity of RSGO enhances the searching ability. This sharing and divergence strategy which adjusts the two tropisms is significant since it allows the root system to adaptively improve its foraging behaviour. At the intial phase of the simulation, the root tip begins to explore the search space. In this way, the main root costs a little time before looking for containing the global optimal region because gravity mentioned in RSGO enhances the exploration ability. Otherwise, by the hydrotropic law of RSGO, the main root would decrease the searching speed near the best solution and increase the number of branches in order to pursue more accurate results.

4.2 Auxin-controlling life-cycle

In this section, we mainly focus on two things in the proposed RSGO model, the auxin regulation dynamic activity underlying root tip branching, and the death of the root tip. From the experiment results we can find the changing process of the population size with the control of the anxin both spatially and temporally. The time and space branching processes based on RSGO under a Rosenbrock-defined environment is shown in Fig. 3 and the process of the root tips population size evolution along with the generations is shown in Fig. 4. The process of the root tip population size variation, just as shown in both Fig. 3 and Fig. 4, can be separated into three phases (branch, decay, and death) which correlate to the life-cycle of all types of plant roots in nature.

Figure 3

RSGO-based spatial and temporal branching with Rosenbrock-defined environment.

Figure 4

The population size evolution process on Rosenbrock environment.

The regulation mechanism of auxin is described in detail in Fig.5-7, which describes the dynamic change of every root tip auxin chroma during the growth of plant roots. We can see clearly from the three phases that if the space region of plant root tip near the local or global best, these will have a higher concentration of apical auxin to multiplying the root number; when the tip enters the plateau region, the apical branching process will stop; when their auxin level is reduced to zero, the tips will be decay. This means that from the resource allocation point of view, when the root system is in an appropriate soil region to produce new root plasticity, but the resource is no longer available, they should be able to shed roots in the resource-poor area.

Figure 5

Auxinoncentration distribution on Sphere.

Figure 6

Auxin concentration distribution on Rosenbrock.

Figure 7

Auxin concentration distribution on Griewank.

4.3 Root system growth

Modeling the plant root architecture can help us relate the knowledge acquired at the root level to the entire root system. Theoretically, we can model the plant root system architecture in different ways, depending on three factors: the objectives, practical knowledge, and parameterization of the different processes available. In this paper, the root system is represented by the development of the root system. This creates a three-dimensional set of connected branch points representing the root and their tips. Each characteristic property in the set is defined in the RSGO model, including age, root type, angle, length, and foraging capacity.

The root-soil interaction and dynamic processes are simulated in the RSGO model under the 3D Rosenbrock, Rastrigin and Griewank respectively (Figure 8–10). From the results, we can see that the root system structure can be flexibly changed according to soil conditions. This flexibility is, in the final analysis, due to the modular structure of the roots, which enables the plant roots to be stretched in regions or patches rich in moisture or nutrients. Although the relationship between root foraging precision and size is still unclear, we can clearly see from the three simulation results that the root system can distribute more of the new root growth to the nutrient-rich region. Few new roots extend to the nutrient-poor areas. That is, the RSGO rules enable the root system place their new roots with more precision.

Figure 8

Simulation of the root system structure with the Rosenbrock-defined environment interaction.

Figure 9

Simulation of the root system structure with the Rastrigin-defined environment interaction.

Figure 10

Simulation of the structure of root system with the Griewank-defined environment interaction.

5 Conclusions

This paper introduces a new modeling method for plant foraging behaviors from an optimal foraging theory perspective. We show that optimal plant foot foraging and auxin-controlled root growth are related RSGO, the optimization method imitating the behavior of the plant root growing, was used to solve the distributed optimization problem. In order to verify the new algorithm, several simulations were performed with representative benchmark functions, followed by a brief analysis and discussion on the implications of tropic root growth, auxin-controlled population dynamics, and root structure simulation. Analysis of the simulation results verify these characteristics, supporting the use of RSGO as an algorithm to solve complex real-world problems.

Acknowledgements

This research is partially supported by National Natural Science Foundation of China under Grant 61602343, 51607122, 61305082, 51575158 and 51378350.

Conflict of interest: The authors report no potential conflicts of interest in this work and have nothing to disclose.

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Received: 2016-5-14

Accepted: 2016-9-14

Published Online: 2016-12-15

Published in Print: 2016-1-1

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