Open Life Sci. 2016; 11: 447–457
Special Issue on CleanWAS 2015
Open Access
Weixing Su, Lin Na, Fang Liu, Wei Liu, Muhammad Aqeel Ashraf, Hanning Chen*
Artificial Plant Root System Growth for Distributed
Optimization: Models and Emergent Behaviors
DOI 10.1515/biol-2016-0059
Received May 14, 2016; accepted September 14, 2016
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
*Corresponding author: Hanning Chen, School of Computer Science
and Software, Tianjin Polytechnic University, Tianjin, 010038, China,
E-mail: [email protected]
Lin Na, Fang Liu, School of Computer Science and Software, Tianjin
Polytechnic University, Tianjin, 010038, China
Wei Liu, College of Information and Technology, Jilin Normal University, Siping, 136000, China
Muhammad Aqeel Ashraf, Faculty of Science and Natural Resources,
University Malaysia Sabah 88400 Kota Kinabalu, Sabah, Malaysia
International Water, Air & Soil Conservation Society 59200 Kuala
Lumpur, Malaysia
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
© 2016 Weixing Su et al., published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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448 W. Su, et al.
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 rootforaging, 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.
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Artificial Plant Root System Growth for Distributed Optimization: Models and Emergent Behaviors In the RSGO model, the plant root system is defined as
consisting of several root tips. Mathematical expressions
can be expressed as follows:
t
t
(1)
=
RS
|i
{θ=
i
1, 2, ,=
P ; t 1, 2, , T }
Where
(2)
θ = xit , fi t , nit , α it , φit
t
i
Here θ denotes a root tip; P denotes the count of
the root tips at time t and T denotes the final time of the
t
root system growing process. The root tip θ i exhibits five
t
t
characteristic parameters: space position xi , fitness fi ,
t
t
t
t
nutrient ni , the auxin α i and orientation φi . α i depends
t
t
upon fi and ni which control the tip’s activity, foraging
(elongate), branchin, or death at time
t. If foraging, the
t
φi
root tip moves with an orientation
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 xi = ( xi1 , xi 2 ,... xiD ) and φi = (φi1 , φi 2 ,...φi ( D −1) ) ,
respectively. Here xid ∈ [ld , 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:
t
i
t
 nt + 1 if fi t +1 < fi t
nit +1 =  it
else
ni − 1
(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.
t
Then the auxin concentration α i , which combines the
health and energy states of the ith root tip, is manipulated
according to the following equations:
t
f t − f worst
healthi = ti
t
fbest − f worst
(4)
t
energyi t =
=
α it ξ
t
nit − nworst
t
t
nbest
− nworst
healthi t
Pt
∑ health
(5)
+ (1 − ξ )
t
energyi t
Pt
∑ energy
j
=j 1 =j 1
t
j
t
, ξ ⊂ [0,1] (6)
t
t
t and n
where f worst
/ f best
worst / nbest are the current worst/
best fitness and nutrient of the whole root system at time t.
449
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:
(7)
t
S=
P t × Cr
m
t
where S m denotes the group size of main root being
selected, Pt is the entire number of root tips, and Cr is the
t
selection probability. Lateral roots are calculated by S l
t
=Pt- S m .
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:
t
(8)

branching

nobranching
> BranchG
if α i
otherelse
t
If θ i is chosen as a main root and its auxin
concentration is adequate to implement branching, the
t
branching number wi of θ i is determined by:
(9)
S t  R α t (S − S ) + S 
=
i

1 i
max
min
min

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 θ it 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
t
θ i ) is randomly falling within one of these subzones.
Then these new branching tips will grow as:
φ tj +=1 φit + λ jφmax / Smax
xtj+1 =
xit + R2lmax H (φ tj +1 )
(10)
(11)
t
where j ⊆ [ S min , S i ] is the root branch index of
t
root tip θ i , λ j ⊆ [1, S max ] is the selecting subzone
t +1
number of the root branch x j , φmax is the maximum
growing turning angle, which is limited to π, R2 is random
value between 0 and 1, lmax is the maximum of root
t +1
t +1 t +1
t +1
D
elongation length, and H( φ j ) = ( h j1 , h j 2 , … , h jD ) ∈isR
a Polar to Cartesian coordinates transformation function,
which can be calculated as:
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450 W. Su, et al.
3.5 Root tip death
D −1
htj1+1 = ∏ cos(φ tjp+1 )
p =1
D −1
htjk+1 = sin(φ tj +( k1−1) )∏ cos(φ tjp+1 )
(12)
p= j
+1
1
htjD
= sin(φ tj +( D
−1) )
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:
t
xit +1 =
xit + R3 ( xbest
− xit )
(13)
t
where i ⊆ [1, S m / 2 ] , R3 is a random value in the range
t
(0, 1), and xbest 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 + R4lmax H (φit ) if xit > xit −1
t
(14)
t
where i ⊆ [ S m / 2, S m ] , lmax is the maximum of root
elongation length, and R4 is random value in the range
(0, 1).
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.
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
xit +1 =
xit + R6lmax H (φit +1 )
t
(15)
(16)
where i ⊆ [0, S l ] , R5 and R6 are random values
and the range of values is (0, 1), φmax is the maximum
growing turning angle, and lmax is the maximum of root
elongation length.
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
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Artificial Plant Root System Growth for Distributed Optimization: Models and Emergent Behaviors 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.
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
451
a
b
c
Fig. 1. Simulation on 2-dimensional Sphere considering: (a) hydrotropism; (b) gravitropism; (c) hydrotropism and gravitropism.
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452 W. Su, et al.
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.
a
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 tothe life-cycle
of all types of plant roots in nature.
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, butthe resource is
no longer available, they should be able to shed roots in
the resource-poor area.
b
c
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
Fig. 2. Simulation on 2-dimensional Griewang considering: (a) hydrotropism; (b) gravitropism; (c) hydrotropism and gravitropism.
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Artificial Plant Root System Growth for Distributed Optimization: Models and Emergent Behaviors 453
Fig. 3. RSGO-based spatial and temporal branching with Rosenbrock-defined environment.
Fig. 4. The population size evolution process on Rosenbrock environment.
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454 W. Su, et al.
Fig. 5. Auxinoncentration distribution on Sphere.
Fig. 6. Auxin concentration distribution on Rosenbrock.
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Artificial Plant Root System Growth for Distributed Optimization: Models and Emergent Behaviors 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.
455
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
Fig. 7. Auxin concentration distribution on Griewank.
Fig. 8. Simulation of the root system structure with the Rosenbrock-defined environment interaction.
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456 W. Su, et al.
Fig. 9. Simulation of the root system structure with the Rastrigin-defined environment interaction.
Fig. 10. Simulation of the structure of root system with the Griewank-defined environment interaction.
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 nutrientrich 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.
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
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Artificial Plant Root System Growth for Distributed Optimization: Models and Emergent Behaviors 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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