Download Towards mechanistic models of plant organ growth

Journal of Experimental Botany, Vol. 63, No. 9, pp. 3325–3337, 2012
doi:10.1093/jxb/ers037 Advance Access publication 27 February, 2012
REVIEW PAPER
Towards mechanistic models of plant organ growth
Dirk De Vos1,2,*, Abdiravuf Dzhurakhalov1,2,*, Delphine Draelants2,*, Irissa Bogaerts1,*, Shweta Kalve1,*,
Els Prinsen1, Kris Vissenberg1, Wim Vanroose2, Jan Broeckhove2 and Gerrit T. S. Beemster1,†
1
2
Department of Biology, University of Antwerp, Belgium
Department of Mathematics and Computer Science, University of Antwerp, Belgium
* These authors contributed equally and should be considered joint first authors.
To whom correspondence should be addressed. E-mail: [email protected]
y
Received 17 October 2011; Revised 23 January 2012; Accepted 23 January 2012
Abstract
Modelling and simulation are increasingly used as tools in the study of plant growth and developmental processes.
By formulating experimentally obtained knowledge as a system of interacting mathematical equations, it becomes
feasible for biologists to gain a mechanistic understanding of the complex behaviour of biological systems. In this
review, the modelling tools that are currently available and the progress that has been made to model plant
development, based on experimental knowledge, are described. In terms of implementation, it is argued that, for the
modelling of plant organ growth, the cellular level should form the cornerstone. It integrates the output of molecular
regulatory networks to two processes, cell division and cell expansion, that drive growth and development of the
organ. In turn, these cellular processes are controlled at the molecular level by hormone signalling. Therefore,
combining a cellular modelling framework with regulatory modules for the regulation of cell division, expansion, and
hormone signalling could form the basis of a functional organ growth simulation model. The current state of
progress towards this aim is that the regulation of the cell cycle and hormone transport have been modelled
extensively and these modules could be integrated. However, much less progress has been made on the modelling
of cell expansion, which urgently needs to be addressed. A limitation of the current generation models is that they
are largely qualitative. The possibilities to characterize existing and future models more quantitatively will be
discussed. Together with experimental methods to measure crucial model parameters, these modelling techniques
provide a basis to develop a Systems Biology approach to gain a fundamental insight into the relationship between
gene function and whole organ behaviour.
Key words: Cell division, cell expansion, hormones, modelling, Systems Biology.
Why do we need modelling?
With the advent of the ‘genomics era’, following the
sequencing of the Arabidopsis genome (TAGI, 2000), plant
biologists are witnessing an explosion of genomes that are
being sequenced. One way to view these genome sequences
is as a ‘parts list’ for building functional plants. Therefore,
the challenge for molecular plant biologists is to understand
how the tens of thousands of genes encoded in each of these
genomes work together. The complex regulatory networks
formed by these genes result in highly reproducible growth
and developmental patterns in the function of the environmental conditions encountered. Genomics approaches
are now extensively used to map the organization of
genome-wide genetic networks into sets of interacting
functional modules. However, understanding and predicting
how these networks ultimately control growth and development is still extremely challenging, if not impossible.
Inversely, classical developmental (molecular) genetics approaches have focused for years on key regulatory modules
by meticulous functional characterization of the genes
involved, through monitoring their behaviour, by altering
their expression or function genetically, and/or by administering chemical perturbations. These approaches have
yielded a treasure of information on dozens of regulatory
modules involved in different aspects of plant development.
ª The Author [2012]. Published by Oxford University Press [on behalf of the Society for Experimental Biology]. All rights reserved.
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3326 | De Vos et al.
With the exception of the interaction between hormonal
pathways, the functional interaction between various regulatory pathways is hardly ever studied. Nevertheless, most
of these pathways ultimately affect growth and development
and therefore somehow are all part of the complete
regulatory circuitry of the plant.
The conclusion from the above is that there is a clear need
for a new approach that integrates the knowledge obtained
from genome-wide and molecular genetics studies in order
to come to a mechanistic understanding of plant growth
regulation. This approach will have to be able to cope with,
on the one hand, the sheer number of genes involved and,
on the other hand, the frequently non-linear interactions
resulting from complex interactions such as feedback and
feed-forward loops, competition for substrates and substrate saturation. These complications make it impossible
for the human brain to understand the global network fully.
Moreover, the genetic networks are intertwined at the
molecular level with the entire complexity of plant metabolites, but also at the higher level with cell–cell and organ–
organ interactions (Beemster et al., 2003), so the approaches
to integrate our experimentally obtained knowledge need to
be multi-scale.
One approach to start tackling this challenge is to build
computer simulation models ‘from the bottom-up’, essentially reverse-engineering the plant by focusing on those
modules that most directly link to growth and extend from
there. This task involves converting the current knowledge
of regulatory processes into sets of mathematical equations
and testing these to see if, together, they lead to realistic
behaviour. When they don’t lead to realistic behaviour, this
indicates gaps in our knowledge. This then needs to be
addressed experimentally, creating a Systems Biology feedback loop between experiments and adaptation of the
models (Kitano, 2002). In this contribution, the approaches
and tools that are currently available to experimental biologists to achieve this ambitious goal will be looked at.
Furthermore, the progress that has already been made in
modelling several key regulatory processes, what the limitations of the current models are, and the possible solutions to
overcome these will be discussed.
The cell as the central unit of plant growth
Essential for building a model is determining its scope. As
indicated, plant growth regulation is a multi-scale process,
encompassing molecular, cellular, tissue/organ, and wholeplant levels. Currently, it may be too ambitious to cover
this entire spectrum, but it seems feasible to address the
molecule to organ scale as a first step. Given that many
developmental studies focus on single organs, this correlates
closely with the available knowledge. It is clear that, in this
context, the unit of the plant cell plays a crucial role. Not
only does it form an organizational level between molecule
and organ, but cells can also be seen as basic units of the
plant that semi-autonomously interact with their direct environment and compartmentalize the molecular interaction
networks that govern their development. This perspective is
in line with the evolution of higher plants from unicellular
organisms. Moreover, in terms of developmental processes,
the complexity at the cellular level is relatively limited. Cells
can grow, divide, die, or retain their current size. In contrast
to animal systems, migration does not play a significant
role in plants due to the presence of the cell wall. To start
building mechanistic models of organ growth it follows that
at least three regulatory modules need to be implemented:
a cell cycle module that regulates cell division, a cell
expansion module, and a signalling module that spatially
co-ordinates these processes at the organ scale. It is not
coincidental that models for each of these modules have
separately been developed over the past few years. These
can serve as a basis for modules to include in whole organ
models.
Modelling frameworks
A modelling framework makes it possible to formulate the
‘biological rules’ that describe the interactions involved in
the various regulatory modules, usually in terms of mathematical equations. In addition, it provides a simulator capable of executing these rules and producing visual and/or
numerical output. To this end, a number frameworks have
been developed that are suitable for simulating multicellular plant systems. An overview of them is listed in
Table 1. Each of these frameworks has its strengths and
weaknesses and it is therefore useful for the aspiring
Systems Biologist to review them here.
L-studio and Virtual Laboratory (VLAB) are based on an
underlying fractal generator with continuous parameters
(CPFG). L-studio is available on Windows and its counterpart VLAB on UNIX/Linux. Development of VLAB has
apparently stopped about a decade ago. L-studio supports
descriptive and mechanistic models for plant development.
Distributed since 1999 (Karwowski and Prusinkiewicz, 2004)
the simulator is originally based on L-systems (Lindenmayer,
1975) and has since been extended to vertex-vertex (vv)systems (Smith, 2006). They can be used to model plant
morphology and plant growth, in which plants are described as an arrangement of functional modules such as
buds or leaves. L-studio can also include the interaction
between plants and their environment, for example, for
simulating the diffusive transport of water in the soil, for
calculating the distribution of light in a plant canopy, etc.
L-studio is not an open source project. It is distributed
in binary format, including extensive documentation and
sample models. An evaluation version, as well as the price
and licensing information for the full version are available
upon request.
OpenAlea (Pradal et al., 2008) is a modelling and simulation framework in plant ecophysiology. It consists of a set
of components to analyse, visualize, and model the functioning and growth of plant architecture, implemented in the
Python programming language (Chun, 2001). A key goal
of this project is the capability to exchange, integrate, and
Mechanistic modelling of organ growth | 3327
Table 1. Simulators for plant growth modelling.
Simulator
URL
Implementation
Application examples
L-studio
VLAB
OpenAlea
GreenLab
CellModeller
algorithmicbotany.org/lstudio/
algorithmicbotany.org/vlab/
openalea.gforge.inria.fr/dokuwiki/doku.php
liama.ia.ac.cn/wiki/projects:greenscilab:download
www.archiroot.org.uk/doku.php/navigation/
cellmodeller
virtualleaf.googlecode.com
http://gin.univ-mrs.fr/GINsim
http://www.copasi.org
C++, OpenGL
C++, OpenGL, Qt
Python, Qt
C, TCL/TK
Python + C++,
OpenGL
C++, Qt
Java
C++, Qt
Demonstration models: plant structures, plant ecosystems
Demonstration models: plant structures, plant ecosystems
Light interception efficiency in leaf shape
Demonstration of Corner, Roux, and Leeuwenberg models
Choleochaete morphology, trichome patterning, branch outgrowth
VirtualLeaf
GINsim
COPASI
re-use existing tools for the rapid development of models.
The use of Python, an interpreted language, for its implementation facilitates this and also ensures that it can be
deployed on all platforms. It also provides a user-friendly
environment with an interface to the inner structure of its
plant models. Elementary programming skills (Python) are
needed to formulate new models in OpenAlea (as with
CellModeller and VirtualLeaf) but a visual programming
environment is included (Visualea) to support high-level
model design.
GreenLab (Cournède et al., 2006) is a functional–structural plant model for the fast construction of plants and for
the calculation of yields in terms of organs in environmental
applications concerning the rationalization of agriculture
and forest exploitation. Note that the structural model
is more aimed at organogenesis or morphogenesis studies,
while the functional model is more geared towards ecophysiology studies. GreenLab, with the GreenSciLab, tool can
simulate different botanical plant architectures, plant growth
processes, the effects of climate conditions on plant growth,
and calibrate the model from experimental data on real
plants. This tool has a user-friendly interface and visual
output. It is well documented and freely available.
CellModeller is an environment for simulating twodimensional multicellular models of plant tissues where the
mechanical properties of the cell wall have been taken into
account (Dupuy et al., 2008). The plant is defined by
various structural entities at different scales represented in
different levels of the model: vertices, walls, cells, tissues,
and organs. The relationships between entities in different
levels are defined by vertical functions, while interactions
between units of the same level (i.e. neighbouring relationships) are referred to as horizontal functions. The behaviour
and properties of these entities can be controlled through
user-specified models, using the Python scripting language.
The biomechanical description of cell wall dynamics as cell
wall response to turgor pressure through cell wall viscosity
is well developed in this simulator. It is an open source
project, available on all platforms, and it is distributed
under the GNU General Public License. The distribution
includes documentation and a tutorial.
VirtualLeaf is a recently developed modelling and simulation tool of plant tissue morphogenesis (Merks et al., 2011).
It is a cell-based framework that uses an energy minimiza-
Plant tissue growth, vascular and biochemical patterning
Flowering in Arabidopsis. Yeast cell cycling
Steady state, metabolic control/ sensitivity analyses, time course simulation
tion method for describing the expansion, division, and
shape change of cells, including cell wall mechanics. The cell
walls are flexible as they consist of multiple viscoelastic
elements linked by nodes. A proxy for turgor pressure has
been implemented as a target area that each cell would
reach in the absence of counteracting cell wall forces. The
intracellular diffusion and reaction of chemicals operate
within each cell and cell wall. Ordinary differential equations
are used to describe biochemical and genetic regulatory
networks and diffusive and active transport. VirtualLeaf
defines the biological entities (such as cells, cell walls,
transporter proteins, signalling molecules) as C++ classes.
Models are defined through plugins, written in the C++
programming language (Stroustrup, 2005), using those
classes. VirtualLeaf is an open source project, distributed
under the GNU General Public License and is available on
all platforms. It includes several demonstration models and
a tutorial is available (see Supplementary data in Merks
et al., 2011). Detailed instruction in its usage in the form of
an experimental protocol is also available (Merks and
Guravage, 2012).
Biological regulatory networks allow cells or organs to
adjust their biochemical behaviour to internal and environmental changes. GINsim (Gene Interaction Network simulation) is a simulator for the qualitative modelling and
analysis of genetic regulatory networks (Naldi et al., 2009).
In GINsim the user specifies a model of a genetic regulatory
network in terms of asynchronous, multivalued logical
functions, and analyses its qualitative dynamical behaviour.
A regulatory network is modelled as a regulatory graph,
with nodes representing genes or regulatory products
and with arcs representing the interactions between them.
Graphical settings of genes and interactions can be interactively modified using either default or user-defined parameters. This logical regulatory graph is the input and a
dynamical graph is generated as the output. Modelling
attributes determine the dynamical behaviour of the network. GINsim is well documented, freely available for
academic users and available on all platforms.
COPASI (Complex Pathway SImulator) is a software for
creating and solving mathematical models of biochemical
networks and analysing their dynamics (Hoops et al., 2006).
It can be used both through a graphical and a command
line interface. No knowledge of mathematics is required: the
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mathematical model (differential equations) is automatically
generated from the molecular model (chemical reactions).
A molecular model is implemented in a straightforward
way by defining the chemical reactions in a simple format.
COPASI is capable of performing time-dependent simulations using a deterministic or stochastic framework, depending on the user preference. It is equipped with a number of
diverse optimization algorithms that can be used to
minimize or maximize any variable of the model, as well as
for estimating parameter values that best fit a set of data
provided by the user. A number of practical examples such
as steady state and time-course simulations, stoichiometric
analyses, sensitivity analysis (including metabolic control
analysis), are provided (Mendes et al., 2009). COPASI is
a very well documented open source, platform-independent
and user-friendly software.
This overview illustrates the activity in the development
of simulation frameworks suitable for modelling various
aspects of plant development. Most of these simulators aim
to make the modelling accessible to experimental biologists
with a basic understanding of computer programming and
mathematics. They do so by providing a framework that
takes care of all basic simulator functions (input/output,
visualization, solving mathematical equations) and provides
building blocks, at the highest level of abstraction possible
in the framework, to formulate models. Nevertheless, at
present, most studies using these tools have involved mathematicians and computer scientists that directly or indirectly
interact with experimental biologists.
changeable regulatory cyclins that determine the substrate
specificity (Dudits et al., 2011; Inzé and De Veylder, 2006).
The target proteins then facilitate the different transitions
(G1/S or ‘Start’, G2/M, intra-M or ‘Finish’).
In addition to a generally conserved basis of cell-cycle
control, the plant cell cycle has a unique feature in which
inhibitory phosphorylation of CDK (Nurse, 1990) plays
a less prominent role. CDC25 does not seem to play a role
in phosphoregulation at G2/M (Dissmeyer et al., 2010).
Inversely the role of cyclin-dependent kinase inhibitors
(CKIs), that regulate CDKs by binding and thereby
competing with other interacting factors, appear to play
a more central role, possibly replacing the CDC25 function
(Verkest et al., 2005; Churchman et al., 2006; Peres et al.,
2007; Dissmeyer et al., 2009). Their levels also appear to
be controlled by ubiquitin-mediated proteolysis (Marrocco
et al., 2010). Moreover, a relatively unique feature of the
plant cell cycle is that the G2 and M phases can simply
be bypassed. This results in endoreduplication, i.e. consecutive DNA replications without any cell division, leading to
an increased ploidy (Breuer et al., 2010).
The dynamical properties of the general eukaryotic cell
cycle have been successfully modelled around a number of
recurring structural motifs (Fig. 1). Double positive or
negative feedback loops can act like switches (Tyson et al.,
2003). This regulatory motif allows a system to change its
state in a discontinuous way in response to an input signal.
Depending on the direction of the change in signal, the
Cell cycle
The cell cycle is a central module in the context of multiscale modelling of growth and development of higher plants
as it drives the increase of cell numbers required to sustain
organ growth. Essentially, the eukaryotic cell cycle is responsible for balancing the cell’s ‘growth cycle’ in which it
increases its size, with the duplication of DNA and
redistribution of this genetic information to the newly
formed daughter cells. Rather than an endless repetition of
cycling through G1, S, G2, and M phases, cell cycle activity
in plants is flexible. Duration of the cycle can vary widely
depending on genetic background and environmental conditions. This phenomenon shows that cell growth and
division are not strictly coupled and that the cell cycle is
composed of different, and to some extent independent, submodules. A final aspect of cell cycle regulation is that, at
some point, cells exit the cycling activity to start differentiation. The occurrence of distinct cell cycle states in the context
of growth and development implies that the transitions
between successive cell cycle phases (checkpoints) should be
considered as regulatory (sub-)modules. In eukaryotic cells
these cell cycle regulatory modules are strongly conserved.
They are all, to some extent, concerned with regulating cyclin
dependent Ser/Thr kinases (CDK) activity. Indeed, the
central cell cycle ‘engine’ consists of a catalytic CDK-subunit
that phosphorylates different substrate proteins, with ex-
Fig. 1. Basic eukaryotic cell cycle model with regulatory motifs.
The principal regulatory interactions as found in generic cell cycle
models (Csikasz-Nagy et al., 2006) with a central CDK:Cyclin
complex. Solid arrows represent 1 or more reactions (e.g. protein
expression is indicated by arrows pointing to Cyclin and CKI),
dashed arrows represent regulatory interactions. The CDK:Cyclin
activity is regulated by temporal cyclin degradation (via APC and
CDC20), reversible binding to an inhibitor (CKI), and (de-)phosphorylation by an inhibitory kinase (WEE) or an activatory
phosphatase (CDC25). These three mechanism all include feedback from the active CDK:Cyclin complex as illustrated in the
boxes alongside the corresponding regulatory network motifs
discussed in the text (pointed and T shaped arrows indicating
positive and negative influences, respectively).
Mechanistic modelling of organ growth | 3329
transition will occur at a different input value (hysteresis).
Bistable behaviour of mitosis-promoting factor (MPF; i.e.
the active CDK–CYCB complex) can work as a two-way
discontinuous toggle switch (Novak and Tyson, 1993).
This mechanism has been shown to govern start (G1/S)
and finish (intra M) transitions in budding yeast cells
(Cross et al., 2002). In models for cell cycle control
in fission yeast and generic models the intra-M transition
module is proposed to be an oscillator, which alternates
in a continuous way between two states and is typically
based on a negative-feedback loop: CDK-CYCB activating the APC, which activates CDC20, which subsequently
degrades CYCB (Tyson et al., 2003; Csikasz-Nagy et al.,
2006). Importantly, a mathematical model can be built up
from these motifs, possibly with finer molecular detail
added, while conserving the qualitative behaviour of the
respective modules. On top of this regulatory layer, socalled surveillance mechanisms are active that can stop
the cell at the various checkpoints. Depending on the type
of cell and the conditions, this can be size (so the growth
and cell cycle can be balanced), DNA damage and repair,
completion of DNA replication, and convergence of replicated chromosomes to the metaphase plane (Kastan and
Bartek, 2004). In model simulations, this is often represented
by changes in the parameters that govern the system’s
dynamics. As an example, Zwolak et al. (2009) proposed
that in frog egg cell extracts, unreplicated DNA would halt
G2/M progression by activating the inhibitory Wee1 kinase
and inhibiting the activatory CDC25 phosphatase. In the
yeast model by Tyson and Novak (2001) the cell mass
governs the reaction kinetics in various ways to act as
a ‘sizer’ mechanism i.e. cell cycle progression is dependent
on reaching a critical size.
In contrast to general cell cycle behaviour, relatively
limited progress has been made in the computational modelling of the plant cell cycle. This may be attributed to the fact
that there has been somewhat of a lag in the characterization
of the molecular reaction network. Beemster et al. (2006)
have proposed a computational model for cell cycle regulation in Arabidopsis in the context of leaf development. SIMplex (Vercruysse and Kuiper, 2005), a dedicated software
tool (http://www.psb.ugent.be/cbd/papers/sim-plex/), was
used to produce the model that consists of piece-wise linear
differential equations governed by a set of logical rules. A
growth factor drives synthesis of specific cyclins and
subsequent formation of S-phase and M-phase promoting
factors (SPF and MPF, respectively). Despite its simplicity, it
qualitatively captures the behaviour of a proliferating cell,
even reproducing the entry into the endocycle. This approach
combines elements of Boolean networks (composed only of
rules, with discrete state variables) and differential equation (DE) models (composed of rate equations, with
continuous state variables). The strategy of using hybrid
models (consisting in part of an automaton) to overcome the
lack of specific kinetic information has been applied before in
cell cycle modelling (Novak and Tyson, 1995; Singhania
et al., 2011). Tyson and Novak (2001) demonstrated that
reasonable models of the control system in yeast cells, frog
eggs, and cultured mammalian cells can be obtained by
adding specific pieces to a simple generic model based on the
antagonistic interactions between CDKs and the APC. These
models are all built upon two major irreversible transitions
(Start and Finish) between two stable states (G1 and S-G2M) of the dynamical system. Dissmeyer et al. (2009, 2010)
have modified this DE-based generic cell cycle model by
replacing the classical G2/M module based on inhibitory
phosphorylation by two alternatives. The first mechanism
relies on activation by MPF of an alternative M-phase cyclin
and the second mechanism on a G2-phase-specific CDK
inhibitor (itself feedback inhibited by CDK). Simulations
with these model versions have demonstrated that these
mechanisms can produce a distinct G2 phase as required for
the plant cell cycle. Importantly, the above modelling
studies have shown that, especially for intricately wired
networks such as the plant cell cycle, they are indispensable
to understand and predict a system’s behaviour.
Given their sessile life style, growth and development
of plants is, in comparison with animals, more influenced
by the environment. Not surprisingly, the list of factors
involved in cell cycle regulation is long. Auxin and cytokinins are considered indispensable for plant cell cycle progression (Hartig and Beck, 2006), whereas other hormones
play a leading part in more specific conditions (Del Pozo
et al., 2005). Besides plant hormones, many factors play
a role in cell cycle regulation, for example, sucrose, light,
wounding, and ROS (Dudits et al., 2011). It is evident that,
to obtain a better understanding of how cell cycle regulation
affects the patterns of growth and development and vice
versa, these factors should be integrated into future cell
cycle models, eventually leading to powerful, multi-scale,
spatio-temporal models.
Cell expansion
Final plant size and shape are determined by the production
of cells and the subsequent, often anisotropic, expansion of
these cells during, and especially after, they have ceased to
divide. In contrast to cell cycle regulation, the presence of
the cell wall on the one hand and vacuoles that determine
the major volume of the mature cell on the other hand,
make the regulation of the cell-expansion process unique to
plants.
Early scientists like Julius Sachs (1882) already described
that when parenchyma cells expand in a growing root, their
morphological changes were correlated with the volume
increase of the central vacuole. Cell turgor and water uptake
were discussed as being instrumental in causing expansion,
concomitant with stretching of the walls.
Up to now, however, an adequate and generally accepted
molecular model of the cell wall structure and how this
structure allows both an increase in surface and an incorporation of new wall material still remains elusive. Most
models only differ in the types of interaction and spatial
arrangement of the different components (Cosgrove, 2000).
One widely used model states that plant primary cell walls
3330 | De Vos et al.
are a composite material consisting of cellulose microfibrils
embedded in a highly hydrated pectin matrix and tethered
by hemicelluloses (Carpita and Gibeaut, 1993). The specific
spatial arrangements of the load-bearing components make
cell walls that can be mechanically highly anisotropic and
different classes of cell wall-modifying enzymes can regulate
wall behaviour at different developmental stages (Cosgrove,
2005). Furthermore, it is generally accepted that microtubules serve as guides for the cellulose synthase complexes
and that, by doing so, they also influence the mechanical
anisotropy of the cell walls (Paredez et al., 2006).
In contrast to the situation in the regulation of the cell
cycle, models of regulation of cell expansion focus largely
on physical properties, rather than molecular interactions
controlling them. An important first step in the modelling
of cell expansion was the summary of experimental data on
wall extensibility by Lockhart (1965) in a rather straightforward formula that has continued life as the ‘Lockhart
equation’:
r ¼ uðP Y Þ
where r is growth rate, u is extensibility of the cell wall, P is
turgor pressure (the source of cell wall stress), and Y represents the yield threshold (the minimum pressure required
for growth). This equation clearly states that the rate of
cell expansion correlates with a difference between turgorpressure and wall stress threshold, and with the extensibility of the wall, which is dependent on arrangements of
components and even on the location in the organ
(Crowell et al., 2011). This emphasizes that key players of
the cell elongation process, potentially capable of changing
any of these parameters, are to be found in the symplast as
well as in the apoplast. Undoubtedly different sets of genes
and proteins must be expressed at different stages of the
expansion process and/or rendered active or inactive.
Several of these crucial genes and proteins are starting to
emerge, but the complete picture is far from clear. Links with
hormone metabolism and transport complicate the picture
even further.
Wall loosening proteins such as expansins and xyloglucan
endotransglucosylase/hydrolases (XTHs) and other lytic
enzymes are active in expanding cells (McQueen-Mason
et al., 1992; Vissenberg et al., 2000). They exert specific effects
on wall polymers and their interactions, and hence on the
biomechanical properties of the wall in general (Cosgrove,
2005; Van Sandt et al., 2007). To study the molecular basis
of cell wall extension several in vitro techniques can provide
answers to different questions. Measurements of breaking
strength, and elastic and plastic compliance can indicate
changes in cell wall structure. On the other hand, wall stress
relaxation and creep measurements give clues both to
changes in wall structure and to wall-loosening processes
(Cosgrove, 2011) and can be used to monitor the effect of
proteins on cell wall behaviour (Cosgrove, 2005; Van Sandt
et al., 2007). Findings resulting from these approaches can
be used to model changes in cell wall behaviour (yielding
and extensibility) when for example XTHs exert their typical
transglucosylase action on xyloglucan (XET) or when only
hydrolase action (XEH; Nishitani and Vissenberg, 2007) is
present.
Next to the loosening of walls and their turgor-driven
stretching, new cell wall material needs to be laid down to
keep the growth rate high for some time. This can be seen
by the reduction in maximal cell size when for example
cellulose biosynthesis is reduced in a mutant background or
by chemical interference, in different organs as well as in
cell cultures (Arioli et al., 1998; Fagard et al., 2000;
Vissenberg et al., 2000; Lane et al., 2001).
In recent years the first models that simulate the cell
expansion process have appeared. Corson et al. (2009)
modelled changes in cell shape that were observed in the
shoot apical meristem of Arabidopsis upon disruption of cell
wall microfibril orientation by treatment with oryzalin.
Their model, based on a set of ordinary differential equations, describes the elastic behaviour and plastic yielding of
cell walls under turgor pressure generated by solute concentrations in the cells. With this approach, they were able to
simulate the dynamic behaviour of cell growth, resulting
in realistic geometrical properties of cells in normal and
perturbed tissues.
Qian et al. (2010) used a finite element method of a fibrereinforced hyperelastic composite material to model the
mechanical properties of onion epidermis in relation to the
orientation and distribution of the microfibrils in the walls.
They related this model to extensiometry data that show the
relationship between microfibril orientations, stress applied
to the wall and the resulting strain rates (Suslov and
Verbelen, 2006). They developed an inverse modelling
approach that allowed the mechanical parameter set to be
found that leads to a good correlation of the simulated data
with the experimental observations.
After models that capture the essentials of the physical
basis of cell expansion have been developed, a next challenging step will be to incorporate the molecular level regulation.
This will allow linking of these models to upstream regulatory
signals. This is necessary to go from the more or less uniform
tissues that they have been tested on to the spatially and
temporally complex situation in growing organs.
Hormone signalling
The classical plant hormones all affect growth and hence
must directly or indirectly impinge on the regulation of cell
division and cell expansion. Broadly speaking, two functions can be discriminated. The first is the generation of
morphogen gradients, which determine the locations where
cells actively divide and expand. The second, superimposed
function is the signalling of environmental conditions (often
stress) that among many other effects result in an adjustment of the growth rate. Here the focus will only be on the
first function as, for modelling organ growth per se, one
needs some form of morphogen that determines cell cycle
and cell growth activity. Auxins are the best studied plant
hormones in this context. More recently, gibberellins (GA)
have also been shown to be instrumental in controlling the
transition from proliferation to cell expansion (Achard
Mechanistic modelling of organ growth | 3331
et al., 2009; Ubeda-Tomás et al., 2009), but more research is
needed to understand how concentrations of active GA are
linked to this function.
At the level of isolated plant cells, cell division and cell
expansion are mainly controlled by auxins, in collaboration with cytokinins. The auxin:cytokinin ratio represents
an important signal in the formation of cell phenotype and
also in the onset and maintenance of the cell division
process. They appear to influence different phases of the
cell cycle. Auxins exert an effect on DNA replication,
while cytokinins exert control over the events leading to
mitosis (Machakova et al., 2008). Differences in the
concentration of these hormones have indeed been linked
to the spatial occurrence of cell cycle activity in a growing
organ. To establish concentration gradients, active transport plays a crucial role. For auxin, the molecular
components that determine this transport have been
identified as PIN and AUX1 auxin efflux and influx
transporter families, respectively (Paciorek and Friml,
2006). These components form an auxin transport network
that mediates local auxin distribution and triggers different
cellular responses in various developmental contexts.
Auxin function in the cells is mediated by a cascade where
auxin binds to TIR1, the F-box subunit of the ubiquitin
ligase complex SCF(TIR1), thereby stabilizing the interaction between TIR1 and Aux/IAA substrates. This interaction results in Aux/IAA ubiquitination and
subsequent degradation. As a result, the inhibitory action
of Aux/IAA proteins on transcription factor activity of the
Auxin Response Factor (ARF) family is released. These
ARFs then activate auxin responsive elements (AuxREs) that
mediate the auxin-response regulation of genes involved in
many processes among which are cell division and expansion
(Mockaitis and Estelle, 2008). Auxin is known to induce the
expression of some members of the XTH family that modify
cell wall components (Vissenberg et al., 2005). In addition, it
is capable of inducing the acidification of the apoplast, which
can stimulate the activity of cell wall modifying proteins,
thereby inducing cell growth (Kotake et al., 2000).
The knowledge concerning the cell and molecular biology of auxins has formed the basis for several mathematical models that simulate auxin-mediated plant
development (Fig. 2). Mathematical formulation of polar
auxin transport rules, mainly in the form of sets of coupled
differential equations, has led to an impressive array of
models that explain auxin distribution patterns in the
context of many developmental processes in the model
plant, Arabidopsis. Swarup et al. (2005) used such a model
to explain the importance of AUX1 expression in the
epidermal tissue during a gravitropical response. Three
independent papers, published back to back, modelled the
accumulation of auxin at the shoot apical meristem in order
to understand phyllotactic patterning (de Reuille et al.,
2006; Jönsson et al., 2006; Smith et al., 2006). In contrast to
these studies, Stoma et al. (2008) proposed a flux-based
PIN polarization mechanism to explain auxin distribution
at the shoot meristem. Flux-based and concentration-based
mechanisms were combined into a dual polarization model
by Bayer et al. (2009). Grieneissen et al. (2007) modelled
auxin transport in roots based on microscopic observations
of PIN localization to explain the occurrence of an auxin
maximum in the Arabidopsis root tip and, by extension, the
spatial occurrence of cell division and expansion, resulting in
a growing root tip. Further development of this work
resulted in a mechanistic model for lateral root initiation
(Laskowski et al., 2008). Jones et al. (2009) modelled auxin
transport in the epidermis of the root tip to explain the
differentiation of trichoblast and atrichoblast cell files. No
fewer than three different mechanisms were proposed to
describe the role of auxin transport in the formation of leaf
vascular patterns. The first is a reaction–diffusion model
which assumes a single diffusable substance (in contrast to
the classical model that combines the slow diffusion of an
activator molecule with faster diffusion of an inhibitor, that
is transcriptionally stimulated by the activator (Fig. 2)),
while assuming uniform local auxin production (Dimitrov
and Zucker, 2006). The second is a canalization model
(where auxin flow orients the PIN transporters to amplify
transport in that direction further, i.e. down the gradient
transport (Rolland-Lagan and Prusinkiewicz, 2005). The third
is a PIN-mediated transport model where PIN expression is
dependent on auxin concentration and orientation is towards
neighbours with higher concentration (Merks et al., 2007;
Wabnik et al., 2010). Bilsborough et al. (2011) modelled the
interaction between PIN1, auxin, and CUC2, explaining the
formation of serrations at the leaf margin. Lucas et al.
(2008) used stochastic modelling to build an auxin transport-based model of root branching in Arabidopsis thaliana.
All these models have in common that they confirm the
central role of active auxin transport and auxin signalling
in plant morphogenesis. These studies have substantially
expanded our mechanistic understanding of diverse processes like embryonic development, stem cell maintenance,
root and shoot architecture, and tropic growth responses.
However, they also highlight the need to unify the
underlying mechanisms in accordance with the central role
of auxin as a regulator of growth and development. Further
steps in auxin modelling will therefore need to assess the
level of detail, i.e. which processes and how many levels of
organization to incorporate in order to obtain suitable
models for the developmental process under study (ten
Tusscher and Scheres, 2011). However, the usefulness of the
auxin models also implies the need for similar models for
other hormones, particularly cytokinin and gibberellin, in
the context of organ growth and development. However,
currently, there is insufficient knowledge on the localization
of synthesis and on transport mechanisms on these
hormones to construct similar models.
Parameter space exploration: transitions in
the steady-state patterns of hormone
transport
As outlined above, most models describe the biological
system with the help of sets of coupled differential evolution
3332 | De Vos et al.
Fig. 2. Auxin transport models. (A) Classical reaction-diffusion mechanism with slow diffusion of an activator molecule (here auxin)
and faster diffusion of an inhibitor, that is (transcriptionally) stimulated by the activator. The resulting local activation and lateral
inhibition can lead to characteristic leaf venation patterns (cf. Meinhardt et al., 1998). Pointed and T shaped arrows represent
activation and inhibition, respectively. (B) In cellular models of auxin transport, auxin auto-activation corresponds to the activation of
PIN exporters that transport auxin in the direction of higher fluxes (canalization) or higher concentrations. (C) Schematic
representation of plant cells with PIN transporters located in the plasma membrane as well as the endosome. Polar re-distribution of
PIN transporters is proposed to depend on a mechanism that senses the auxin concentration in neighbouring cells (or, for example,
in the apoplast as proposed by Wabnik et al., 2010) or the auxin flux. This signal could feed back on PIN translocation preferentially
retaining PINs at one side of the cell.
equations. Implicitly, these equations contain a range of
parameters that determine the behaviour of the system. For
the majority of these parameters, no experimental variables
are available. Currently, most modelling papers only present
the (realistic) behaviour of the model for a standard set of
parameters. However, once established most models allow
for a more refined evaluation of the model parameters.
Answers to questions such as, ‘Across what range of values
parameter values do the models yield stable solutions?’ and
‘Which of the parameters are crucial for the overall behaviour and which ones are less important?’, provide crucial
information about the model and pose testable hypothesis
for the experimental biologist.
Mathematical analysis techniques have been developed
for this purpose and their usefulness will be illustrated here
in the context of the transport of auxin hormones in a
simple linear file of cells with a static cell geometry, using
the model of Smith et al. (2006). This model consists of
two equations for every cell. One equation models the
evolution of the PIN1 protein and the other equation
describes the evolution of the hormone auxin (IAA). Both
equations are dependent on the concentration of PIN1 and
the concentration of IAA in that cell, but they are also
dependent on those concentrations in the adjacent cells
with, at most, distance 2. This leads to a non-linear coupled
system of evolution equations that can yield a stable pattern
Mechanistic modelling of organ growth | 3333
of equally spaced auxin maxima. This pattern can be
visualized, for example, using reporter genes. The nonlinearity is a consequence of the active transport between
the cells.
The question one could address is how this pattern varies
when model parameters are changing. For most parameters
in the model, only a range of possible values is known and
no particular values have yet been experimentally determined. It is still not clear which role each of the parameters
plays in the pattern formation.
However, if one of the parameter values is changed, the
stability properties of the pattern might change, resulting in
a new pattern or periodic fluctuations of the concentrations
over time. It is therefore important to understand how the
stability properties of the pattern solution depend on the
parameters of the model.
The study of the relationship between the stability of a
solution and the parameters of the corresponding dynamical system is known as bifurcation analysis (Seydel,
1994). Such an analysis identifies the stable and unstable
solutions and the bifurcation points that mark the
transitions between them. At bifurcation points the
number of solutions change. This analysis usually leads
to a bifurcation diagram that highlights the connections
between stable and unstable branches as the parameters
change. This is illustrated here with the bifurcation
results for the model of Smith et al. (2006) applied on
a row of 20 cells with the IAA transport coefficient T
as the varying parameter (Fig. 3). For the other parameters of the model, similar scenarios can be calculated. It
was observed for some of the parameters that there is
a constant solution with an equal concentration in all
cells. This stable solution does not change in time and
peaks are absent. However, when the parameters are
changed beyond a critical threshold, this equilibrium is
suddenly replaced by a stable pattern of peaks or
a periodic motion of the concentration. When the
stable pattern of peaks appears, the transition point is
known as a branch bifurcation (Fig. 3). When a periodic
motion appears, the transition is known as a Hopf
bifurcation.
The above illustration of a bifurcation analysis of the
auxin patterns shows its usefulness to explore the relationship between all possible patterns and the parameters used
to model the transport of hormones. With the help of bifurcation analysis, for each given mathematical model the
‘parameter space’ can be explored and the stable patterns
that can emerge can be identified. It is also possible to compare multiple models that aim to describe the same biological
system. Furthermore the genesis of a given pattern can be
linked to a particular parameter.
For a low-dimensional system of coupled ODEs, such as
a model for a row of a few cells studied in the paper, there
are excellent numerical continuation tools that automatically identify bifurcation points and switch between
branches and produce the bifurcation diagram. Examples
are AUTO (Doedel et al., 1997) and PyDsTool (Clewley
et al., 2004).
Perspectives
The above review has illustrated the growing integration of
modelling in biological research and the developments that
will increase the power of these approaches to dissect the
behaviour of the models and hence the biological system
under study. No doubt these developments will deliver
powerful new tools for a thorough understanding of plant
growth and development.
It is also clear that for this approach to reach its full
potential, computational and mathematical frameworks need
to be developed further, but, equally important, biologists
also need to evaluate the types of experiments and measurements required to support the modelling. Biologists will need
to develop strategies to validate models at underlying organizational levels such as the cell and molecular levels. At the
cellular level, methods to quantify cell division and expansion
accurately, both at high spatial and temporal resolution,
have been developed (Fiorani and Beemster, 2006). Unfortunately, these methods are still rather labour-intensive, but
several steps are currently being automated, so that, in
future, the capacity for such analyses will increase.
At the molecular level, many tools are available to plant
biologists to study various aspects of the models qualitatively. Particularly in the model species, Arabidopsis, a
wide range of reporter lines are available that report the
transcriptional activity of well-studied genes and are
therefore likely to become part of the modelling activity.
In particular, for the study of auxin transport and accumulation, many extremely useful reporter lines and tools
have been developed. For other hormones such tools are
still largely missing, but may become available in the next
few years. Measurements of metabolites, of interactions
of metabolites, as well as of rates of synthesis, decay,
and transport are extremely difficult. Nevertheless, efforts
should be made to develop strategies to measure such
quantities. Further integration of modelling in biological
research will increase the need for such measurements.
The need for measurements of key model parameters is
a first example of how modelling can direct the biological
experiments that need to be developed in order to increase
our understanding of the biological system under study. It
clearly shows how models can direct research to unanswered pertinent questions (Systems Biology). By integrating new detailed knowledge such models can gradually
provide increasingly accurate descriptions of specific developmental processes. Besides this increasing detail, combining different sub-models of parts of the plant in a joint
framework will allow the plant and, ultimately, a crop as a
whole, to be accurately represented. Such knowledge-based
models will ultimately replace phenomenological models
build on a statistical basis that are currently most commonly used in modelling field crops. The long-term
perspective of having models built on the function of
detailed gene networks will enable crops to be engineered
with improved growth characteristics specifically tailored
for high yields of particular organs under a defined set of
environmental conditions.
3334 | De Vos et al.
Fig. 3. Bifurcation analysis of hormone transport in a linear file of 20 cells. (a) The bifurcation diagram that depicts the concentration of
the growth hormone auxin in cell number 5 versus the IAA transport coefficient T. The solid lines indicate the stable solutions and the
dashed lines the unstable solutions. The other plots illustrate the steady-state auxin patterns in all cells for the specific places indicated in
the bifurcation diagram. For small values of T, there is only the constant solution with equal concentrations in all cells shown as the
horizontal line in the diagram. When the parameter T becomes larger, this solution loses its stability at a branch point (BP1). In this
branch point there is an exchange of stability to the other branch, also shown in the diagram. There are two stable parts on this other
solution branch with patterns. The solution patterns in area 2 show one big peak as displayed in (c). This pattern can appear when the
IAA transport coefficient T is large. The other stable pattern shows two small peaks as displayed in (g). This pattern appears in area 4 for
a very limited range of the IAA transport coefficient T.
Mechanistic modelling of organ growth | 3335
Acknowledgements
This work was funded by a GOA research grant, ‘A
Systems Biology Approach of Leaf Morphogenesis’ granted
by the research council of the University of Antwerp and
a return grant from the Belgian Federal Science Policy
Office (BELSPO) to DDV.
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