A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
International Journal of Digital Content Technology and its Applications. Volume 5, Number 1, January 2011
A Self-assembling Approach to Simulation of Phototropism
Hongchun QU1*, Youlan WANG2
Key Laboratory of Network Control and Intelligent Instrument (Ministry of Education)
Chongqing University of Posts and Telecommunications, Chongqing 400065, PR China
[email protected]
2
Chongqing Electric Power College, State Grid, Chongqing 400053, PR China
*1
doi:10.4156/jdcta.vol5.issue1.7
Abstract
In this paper we model the phototropism of virtual plants using a self-assembling approach where
individual organs are modeled as interacting intelligent agents. Each agents possess inbuilt rules that
make them can autonomously respond to environmental stimulation, interact with each other and
deform according to their physiological status. The phototropism of virtual plant is simulated via
agent’s reasoning cycle of producing new organ agent with best relative angles which can maximize
the light interception.
Keywords: Phototropism, Intelligent Agents, Self Assembly, Simulation
1. Introduction
Phototropism is directional growth of plants where the direction of development is determined by
the direction of the light stimulation. Phototropism is the most important one of the many plant
tropisms or movements which responds to external stimuli. Growth towards a light source is a positive
phototropism which is enabled by auxins (i.e., plant hormones that have many functions). In this
respect, auxins are responsible for expelling protons (by activating proton pumps) which decreases pH
in the cells on the dark side of the plant. This acidification of the cell wall region activates enzymes
known as expansins which break bonds in the cell wall structure, making the cell walls less rigid. In
addition, the acidic environment causes disruption of hydrogen bonds in the cellulose that makes up the
cell wall. The decrease in cell wall strength leads cells to swell, exerting the mechanical pressure that
drives phototropic movement [1].
Despite decades of research, involving many noted plant modeling approaches and simulation
systems, the simulation of plant phototropism is far from satisfied. Phototropism models using
traditional approaches are commonly confronted with the problem: how to manually and effectively
design the exact growth grammar for plants with complicated structure and physiological process [2]?
Not to mention that all individual plants are distinct entities exhibiting behavior typical of all complex
organisms [3], e.g. preferential organ placement of nutrients-foraging and light-stimulation, differential
distribution of biomass as consequences of environmental heterogeneity, interactions with other
organisms at their own and higher levels of organization, etc. Obviously, these complex behaviors have
no identifiable centers of tactical, as opposed to strategic, control. Plant intelligence arises in complex,
dynamic systems held in balance by complex cause–effect interactions as regards the internal
physiological process and external environment. Traditional mechanistic approaches, no matter Lsystem or AMAP family can not model this feature effectively [4]. New modeling paradigm, such as
the teleonomic method that can capture natural plant behaviors from simple and ‘‘bottom-up”
perspective without loss of reality provides a good option.
This paper presents a novel simulation model incorporating the teleonomic modeling approach for
plant phototropism in which individual organs are modeled as intelligent agents (Figure 1). Each agents
possess inbuilt rules that make them can autonomously respond to environmental stimulation, interact
with each other and deform according to their physiological status. Reasoning capabilities such as
optimal carbon allocation and maximal light interception are incorporated into intelligent agent. The
growth dynamics and phototropism movements of plant are based on the theoretical foundation of
complex adaptive system, which can help the design of individual-based system less ad hoc and more
likely to produce models of the value for plants modeling [5]. The main new feature of this model is
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A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
International Journal of Digital Content Technology and its Applications. Volume 5, Number 1, January 2011
that it allows real plant phototropism resulted from interactions of individual organ agents, i.e.,
emergence from the self-assembling structure, rather than strict physiological mechanisms.
2. The model
2.1. Plant representation
The plant model used in this work is based on multi-agent system [6]. The main idea of this
approach is to decentralize all the decisions and processes of the whole plant level on several
autonomous entities, the intelligent agents, which are capable of communicating together and sensing
light in virtual environment, instead of on a unique super-entity. Therefore, the structural and
functional aspect of a plant is determined by a set of intelligent agents, representing the plant organs,
which allow the emergence of plant global behaviors by their cooperation and competition.
Figure 1. Schematic description of the organ agent
As schematically described in Figure 1, each organ acts as an independent, self-assembling and
autonomous intelligent agent with a sensor that measures environmental conditions and its internal
physiological status and inputs the data into a recurrent RBF neural network (RRBF-NN) based
physiological status predictor to simulate photosynthesis, respiration and transpiration as a holistic
physiological process. The physiological status predictor can output the physiological status of the next
simulation step to decide the actions of the effector.
From the functional perspective, each of these intelligent agent based organs has their own mineral
and carbon storage with a capacity proportional to its volume determined by the genetic parameters
density. These storages carbon and mineral resources are used for its survival and its growth at each
simulation step. During each stage, an organ receives and stores resources directly from external virtual
environment (e.g., ground minerals or sunlight), or indirectly from other organ agents, and uses them
for its survival, physiological functions and development. The organ is then able to convert carbon and
mineral resources in structural mass for the growth and respiration process or to distribute them to
neighboring organs.
As an intelligent agent, each organ uses both traditional and reactive methods to perform its task of
growth and development in each simulation step. Basically, at every simulation step, a plan is
constructed for the organ agent to produce carbon (if it is mature) by photosynthesis; allocate carbon to
organs (internode, fruit) of the metamer; consume carbon by respiration, growth and generating new
organ; regulate available carbon level by adjusting storage/mobilization ratio; update its available
carbon content due to phloem transportation; die due to meeting the senesce rules or reaching its lifespan. This plan is then executed as a series of behavior rules at each simulation step. Figure 2 shows
this plan by rounded-rectangles linked with solid arrows. Generally, the goal of an organ agent is to
keep balance between carbon production, consumption and storage, as well as find best position of
light interception for new organ that make it can generate carbon as much as possible. This goal make
an organ agent might avoid abortion and live as long as possible. To achieve this goal, two reasoning
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A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
International Journal of Digital Content Technology and its Applications. Volume 5, Number 1, January 2011
processes are introduced to (1) handle the internal carbon allocation and (2) find the best rotation
angles for newly generated metamer to maximize light interception. The latter can help to realize the
phototropism.
To identify the position of each organ agents in the virtual environment, the relative coordinates was
employed in this model. Any two neighboring organs are connected by an instance of the data structure
named connection. For each connection conij(mi, mj), a set of attribute vectors XtopA={ORIij[H,L,U]}
picturing the relative rotation angle from parent organ to its direct children are attached.
Figure 2. Simulation cycle and reasoning procedure of the organ agent. Rounded rectangles
denote behavior rules, rectangles denote stages of two reasoning cycles: (1) find the best
proposal of internal carbon allocation that satisfy organs' carbon demand priority and keep
carbon balance between storage and transport; (2) find the best proposal of rotation angle that
make newly produced organ can hold best position to maximize the light perception.
2.2. Virtual environment
The plant is disposed in a virtual environment [7], defined as a particular agent with the sky voxels.
The environment manages synchronously all the interactions between organ agents, like carbon
transport from neighboring organs, competition for light and physical encumbrance.
In order to simulate plant growth in response to light, the distribution of light in the whole sky of the
virtual environment is set by means of the Firmament submodel [8]. Photosynthetic light density is
divided into a certain number of sectors according to a standard overcast distribution [9]. The model of
light interception presented in this paper deals with direct and diffuse fluxes separately.
With the standard overcast distribution, the sky hemisphere is respectively divided into Di and Da
sectors in both vertical and horizontal directions. These sectors are divided so as to have as equal solid
angle as possible. The number of the horizontal azimuth (Da) defined in this model is the mean number
of sectors contained within the inclination zones. There is a zenith sector at the top of the hemisphere
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A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
International Journal of Digital Content Technology and its Applications. Volume 5, Number 1, January 2011
with its direction pointing directly upward. Therefore, the total number of sectors in the hemisphere is
equal to Di×Da+1. The total area of the upper sky hemisphere is 2π; thus the area of the zenith sector is
equal to 2π / (Di×Da+1). The width of the inclination zone is equal to (π/2 - angle of Da) / Di. The width
of sectors in the same azimuth is the same as the numbers of sectors at each inclination level.
The diffuse light in each sector from the sky is distributed according to the zonal brightness of the
standard overcast [10]:
d ( inc ) 
(6 / 7)(1  2 sin(inc ))
,
2
(1)
where inc is the elevation of the sector located in the sky, and d (inc) is the fraction of diffuse
light from sector inc out of the total diffuse light received by the sky. According to the equation,
the brightness of each sector is exclusively determined by the sector inclination. Each sector has
a specific direction. Both direct and diffuse components reach the growing plant through these
sectors.
2.3. Light interception
The simulations presented in this paper focus on the light interception and corresponding plant
motion, the phototropism. Photosynthesis is the process by which the plants increase their carbon
storage by converting light they received from the virtual environment. Each point of the leaf can
receive light from the sky according to its position and angles respect to the light source in order to
simulate a simple organ movement.
The process of light interception and photosynthesis provide estimates of carbon gain for the
simulated orange tree as a function of climatic parameters and the physiological state of the leaves. The
photosynthetic active radiation Is consists of direct radiation (Idir) and diffuse radiation (Idif) according
to the relationship between measured and potential global radiation [11]:
Is  I dir  I dif ,
(2)
The direct radiation is produced by the solar ray from sun passing directly through the atmosphere
without any scattering. A solar ray is considered a source vector that originates at the solar position
H0(x0, y0, z0) in the upper hemisphere of the virtual environment. Solar position defines the direction of
the direct solar ray. The diffuse radiation is denoted by photon flux scattered in the atmosphere due to
contact with dust and water vapor. Every leaves in metamers can receive diffuse radiation which
scattered uniformly from all directions of the visible hemisphere.
Assume that the centroid of a leaf has the position of H(x, y, z) with azimuth α and inclination β
with respect to the base of organ, giving the incidence of sun light Rdir(θ, φ) with azimuth angle θ and
elevation angle φ, the direct photon flux density Idir can be calculated as a function of the relative
geometry between the solar ray direction and the leaf orientation:
I dir  ( Rdir / sin  )* | cos  *sin   sin   cos  *cos(   ) | ,
(3)
where Rdir is the incident direct radiation, and the relationship between the position of leaf and the solar
position satisfy:
 x   x0 
 sin  / tan  
   


 y    y0     cos  / tan   ,
z z 


1
   0


(4)
where μ is the variable parameter of the equation of the straight line. Leaves in canopy always cast
shadows. In such case, the irradiance following the direction of solar ray is decreased, that is,
multiplied by a shading factor (the probability that a leaf is sunlit) γ computed by Beer’s law:
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A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
International Journal of Digital Content Technology and its Applications. Volume 5, Number 1, January 2011
  exp(V *
h
),
2  sin 
(5)
where V is the total number of leaves in the canopy, h is the height of the orange tree.
Regarding diffuse radiation, the upper sky hemisphere is divided up into m solid angle sectors in
horizontal direction. Each sector i corresponds to directions with elevation φi and azimuth θj. Assuming
that the diffuse incident radiation conforms to an isotropic distribution, and considering the extinction
coefficient e according to Beer’s law:
e
2
1 1
,
*
1  1.6sin  1  1  
(6)
the diffuse photon flux density Idif coming from each sector i can be written:
m
I dir   e *( Rdif / sin i ) *(sin 2 (i   / 2)  sin 2 (i   / 2)) *( /  ) ,
(7)
i 1
where σ is the scattering coefficient of leaves, i.e. the sum of leaf reflectance and transmittance (σ≈0.2
for photosynthetically active radiation), Rdif is the incident diffuse radiation above the simulated tree.
When an organ is ready to birth a new organ, the possible range of relative rotation angle can be
specified. Therefore, the best azimuth and inclination of leaf-blade with respect to the base of new
organ that can maximize the light interception should be calculated as:
 max ,  max  arg max ( I dir  I dif ) ,
(8)
 ~  ,  ~  
where Idir and Idif are respectively the direct and diffuse photon flux density calculated by equation 3
and equation 7. Since the position of each organ is calculated by the relative geometry from its parent
(the absolute position in the global coordinate system is obtained by request to the graphic engine), we
assume that the parent organ has the position H(x,y,z), then the position of the base of the new organ
can be given as H’(x’,y’,z’) with:
(9)
x '  h cos  max cos  max ,
y '  h cos  max sin  max ,
z '  h sin  max ,
(10)
(11)
where h is the length of the parent organ.
3. Simulation
Simulation model using self-assembling approach in this paper provides the architecture of
distributed intelligence for plant growth, i.e., each organ agent can autonomously perceive local light in
the sky hemisphere of the virtual environment (as illustrated in Figure 3). This crucial infrastructure
makes it possible to vividly simulate the shoot phototropism, which might have significant effects on
the light interception, dry matter production and yields of virtual plants [12].
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A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
International Journal of Digital Content Technology and its Applications. Volume 5, Number 1, January 2011
Figure 3. A snapshot of real time simulation of leaf intercepted radiation intensity in sky hemisphere
of the virtual environment. Due to direct and scatter radiation as well as crown distribution, the
aboveground leaves illustrate their actual intercepted light intensity via being marked by different
colors.
Figure 4. Simulated phototropism of plant with structure adaptation (a). Fully growing form with leaf
cover and fruit production of the same plant were given in (b) and (c).
Plants are able to modify their foliage architecture in response to the incident angle of light source
[13]. Typically, phototropic response is dominated by the blue region of the spectrum. This effect is
mediated at least partially by phototropins [14] which can drive the reorientation of leaves at early
ontogenic stages of plants [15]. In this paper, we model plant phototropism by producing new organ
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A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
International Journal of Digital Content Technology and its Applications. Volume 5, Number 1, January 2011
with relative angles (the azimuth and inclination) respect to its parent. Once a bud of an organ agent
accumulated enough carbon and prepared to generate a new organ, the azimuth and inclination angles
related to its parent should be chosen according to which angles can make the new leaf get the
maximum light (equation 8). This process was autonomously controlled by the reasoning cycle 1 of the
organ agent.
Figure 4a shows the simulated phototropism of plant with structure adaptation. The fully growing
form with leaf cover and fruit production of the same plant were also given in Figure 4b and c. In
Figure 4, light was coming from top right corner of the sky hemisphere, so that right-sided leaves and
apical buds gradually found themselves in the shade. Since the carbon allocation and the activation of
the shoot apical buds depends partially on the access to light intensity, only left-sided apical buds
continued to develop. Consequently, the plant adapted to the constraint by developing branches which
are bent downwards. In contrast with our organ movement approach, a phototropism model for
cucumber canopy was developed by Kahlen et al. [16] using a parametric L-system. Their approach
directly modeled the leaf movement induced by gradients (the red to far-red ratio) in the local light
environment of each leaf.
4. Conclusion
In this work we presented a novel approach to simulation of phototropism for virtual plants
integrating technologies of intelligent agent as well as the knowledge of existing functional–structural
plant models, instead of providing a pure and traditional physiological plant model. The architecture of
the whole plant is built by self-assembling organs which are intelligent agents with both functional and
geometrical structure. The development of plant is achieved by the flush growth of organ agents
controlled by their internal physiological status and external environment. The phototropism of virtual
plant is simulated by producing new organ with relative angles: the azimuth and inclination. Once an
organ agent accumulated enough carbon and prepared to generate a new organ, the azimuth and
inclination angles related to the parent organ should be selected according to which angles can make
the new leaf get the maximum light. This process was controlled by the reasoning cycle of the organ
agent. These simple rules and actions executed on the organ level can cause the complex adaptive
behaviors on the whole plant level: adaptation of plant growth to environmental heterogeneity and the
phototropism.
5. Acknowledgements
This research is supported by the National Natural Science Foundation of China (50804061), the
Natural Science Foundation Project of CQ CSTC (CSTC, 2009BB2281).
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A Self-assembling Approach to Simulation of Phototropism
Hongchun QU, Youlan WANG
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