A dynamic root system growth model based on LSystems
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Abstract
Understanding the impact of roots and rhizosphere traits on plant resource efficiency is important, in particular in the light of upcoming shortages of mineral fertilizers and climate change with increasing frequency of droughts. We developed a modular approach to root growth and architecture modelling with a special focus on soil root interactions. The dynamic threedimensional model is based on LSystems, rewriting systems wellknown in plant architecture modelling. We implemented the model in Matlab in a way that simplifies introducing new features as required. Different kinds of tropisms were implemented as stochastic processes that determine the position of the different roots in space. A simulation study was presented for phosphate uptake by a maize root system in a pot experiment. Different sink terms were derived from the root architecture, and the effects of gravitropism and chemotropism were demonstrated. This root system model is an open and flexible tool which can easily be coupled to different kinds of soil models.
Keywords
Root system development Phosphate uptake Tropism LSystem Mathematical modelling Sink termIntroduction
Increasing plant water and nutrient use efficiencies is a major challenge that must be met in order to respond to the rising demand for sufficient food supply of an ever growing human population. Plant productivity is governed by many environmental factors such as radiation interception and water and nutrient availability in soils. Thus, a fundamental understanding of the key processes determining plant growth is necessary in order to improve cropping systems and cultivars for resource limited environments (de Dorlodot et al. 2007). Understanding the impact of roots and rhizosphere traits on plant resource efficiency is important, in particular in the light of sustainable production with reduced fertilizer input, potential shortages and increasing costs of fertilizers and climate change with increasing frequency of droughts. In this context, root architecture is a fundamental aspect of plant productivity (Lynch 1995) and thus needs to be accurately considered when describing root processes.
Existing root system models can be divided into pure root growth models, which focus on describing the root system’s morphology, and more holistic models, which include several root–environment interaction processes, e.g. water and nutrient uptake.
The first descriptive threedimensional root system models were presented by Diggle (1988) (RootMap) and Pagès et al. (1989). These models described a herringbone topology and defined root properties for every topological order in the root system. Based on these ideas, Lynch et al. (1997) and Spek (1997) developed new root growth models (SimRoot and ArtRoot) with a strong focus on visualisation. Pagès et al. (2004) presented a root system model in which the different types of roots composing the root system topology are not strictly related to the topological order (Root Typ).
These dynamic threedimensional root growth models have provided a basis to couple root growth to plant and soil interaction models in order to simulate environmental processes such as water and nutrient uptake. A root system model developed for such a process oriented analysis was presented by Fitter et al. (1991) to study the exploration efficiency of root systems in dependence of root architecture. It could be demonstrated that both root topology and link lengths strongly influence nutrient uptake efficiency (Fitter and Strickland 1991). Dunbabin et al. (2006, 2002) predicted water and nutrient uptake, in the presence of exudates (Dunbabin et al. 2006), by integrating RootMap into a simulation environment. Dunbabin et al. (2004) analysed the effect of different root system architectures on nitrate uptake efficiency. Walk et al. (2006) used SimRoot to assess the trade off effects of different root system morphologies on phosphorus acquisition. Based on the model of Pagès et al. (1989), a model of water conduction within roots has been proposed by Doussan et al. (1998) and combined with a model for water transport in soil (Doussan et al. 2006). Clausnitzer and Hopmans (1994) developed a new root growth model and analysed water flow in the soil root zone which was further extended by Somma et al. (1998) for nutrient uptake. Javaux et al. (2008) combined the model of Somma et al. (1998) and the model of Doussan et al. (1998) to comprehensively describe water flow between the soil and plant root domain. Roose and Fowler (2004) derived sink terms for nutrient and water uptake from a continuous root system growth model which can be solved analytically.
Beside the effects of root growth dynamics and architectural traits on the soil (water and nutrient uptake), a major challenge of root modelling is the dynamic interactions of soil properties on the root growth and architecture itself. Water availability, nutrient concentration as well as mechanical impedance have been shown to strongly influence plant root traits (e.g. Hodge 2004; Eapen et al. 2005; Bengough et al. 2006). Integrating root plasticity in response to the growth environment into root architecture models remains a major challenge for an accurate model based analysis of plant growth strategies in different environments (Tsutsumi et al. 2003). Root responses to resource heterogeneity in the soil are subject to ongoing empirical research concerning phytohormone mediated signalling pathways and species specific morphological and physiological reactions (e.g. Forde and Lorenzo 2001; Callaway et al. 2003; Hodge 2004; Zhang et al. 2007; Peret et al. 2009). This requires a flexible modelling approach for continuous incorporation of new knowledge.
Processes and interactions in the soil–plant–atmosphere continuum are often described by partial differential equations. The most important equations are the Richards equation for water flow (Jury and Horton 2004) and the convection diffusion equation for solute transport in soil (Barber 1995). With increasing computational power, it is now possible to solve these equations numerically in three dimensions using finite difference, finite volume or finite element methods. The complexity of the root–soil system, however, requires an accurate and detailed description not only of each subsystem (e.g. root growth, root architecture, water and solute transport), but also of their mutual linkage and influence. Thus, a modelling approach with a modular structure would be most appropriate to join developments in modelling the individual components of the rhizosphere. The powerful numerical solvers available today (e.g.: Comsol Multiphysics, FlexPDE), combined with techniques such as operator splitting, yield a detailed description of the single subsystems and subsequently enable these to be joined into a comprehensive system analysis.
This study presents such a modular approach for root growth, root architecture and root–soil interactions based on LSystems. LSystems are often used in plant architecture modelling (Prusinkiewicz 2004) and have also been applied to root systems (Prusinkiewicz 1998). We provide an implementation of our Lsystem based root growth and architecture model in Matlab and the integration of the root system description into a plant–soil interaction model. This approach considers the plasticity of root growth and branching strategies as influenced by local soil properties. From the root system architecture model, parameters such as density distributions can be derived and used in soil models. Matlab makes it is easy to link the model to existing simulation codes (e.g.: Java, C, Fortran) or to apply external numerical solvers. The new approach is exemplified in a simulation study of maize root growth and phosphate uptake in a pot experiment. The effects of different sink terms derived from the modelled root architecture and the role of gravitropism and chemotropism is demonstrated. Our main objective is to provide a mathematically sound and publicly accessible dynamic root growth model with a focus on a modular approach to integrate various types of interactions between root architecture and the soil environment.
Model description
Introduction to LSystems
The production rules have to be designed such that the result does not depend on the discretisation of the overall simulation time. The delay rule utilises two different kinds of parameters. The local age t changes when the production rule is applied, while t _{ end } and N _{ s } are predetermined initial values. We use a syntax in which local parameters are given in front of the semicolon and initial parameters are given after the semicolon, i.e., D(t): = D(t ; t _{ end }, N).
The following sections describe the basic production rules of the LSystem model for root growth. A major concern is that all production rules are based on biological mechanisms. Our production rule for elongation of individual roots is based on the continuous root growth model of Roose et al. (2001). If required, it can be replaced by other models.
Axial growth
In this section, we have described a production rule for the growth of an individual root. In the next section, we present an LSystem rule for lateral branching and summarise the root parameters that are needed for a single root with branches.
Lateral branching
In a root system, every root of a certain order produces lateral branches. A root is therefore divided into three zones: the basal and apical zones near the base and the tip of the root, respectively, where no branches are produced, and the branching zone where new roots of successive order are created. The self similar structure of the root system is used in the model description.
The first expression of the rule corresponds to the first expression of Eq. 4 with the additional constraint that growth does not exceed length l _{ s }. The second expression describes the case if the length exceeds l _{ s }. In that case, a segment with the remaining length \(F_{l_sl}\) is produced and the successor, N _{ s } is applied with the correct time overlap t + Δt − (t _{ end } − t _{0}). Otherwise, local time is increased. The following parameters and initial values are needed: S(t,l) : = S ^{Δt }(t,l; t _{0}, t _{ end }, N _{ s },λ,Δx). In this way, a section is described by a number of segments with lengths equal to or less than Δx. Using the section growth rule, the basal zone is succeeded by the branching zone which is then succeeded by the apical zone.
Parameters for a root with lateral branches
Mean  SD^{*}  Description  Unit 

r  r _{ s }  Initial growth speed  [cm/day] 
l _{ b }  l _{ bs }  Length of basal zone  [cm] 
l _{ a }  l _{ as }  Length of apical zone  [cm] 
l _{ n }  l _{ ns }  Spacing between branches  [cm] 
n  n _{ s }  Number of branches  − 
δ  δ _{ s }  Angle between root and predecessor  [degree] 
a  a _{ s }  Radius of the root  [cm] 
C  Colour  [RGB]  
N _{ b }  Successive branch 
Root systems
To include effects that arise from gravitation and soil heterogeneities, more mechanisms have to be included, which are discussed in the following section.
Tropisms and root tip deflection
The previous explanation shows that the root system is described by a large number of segments of length less than or equal to Δx. In front of every segment F _{Δx } is a rotation R which is created when the root axis grows, see Eqs. 4 and 5. In the model, we specify R by two rotations (following Diggle 1988). The root tip axis rotates radially by an angle β and then axially by an angle α, see Fig. 2. In reality, the direction of root growth is influenced by mechanical as well as plant physiological properties. We include this in the model by specifying different ways to choose α and β. In this way, we can simulate root tip response to mechanical soil heterogeneities as well as various types of tropisms like gravitropism, hydrotropism or chemotropism. The specific growth behaviour can be chosen for every root type.
Tropisms
The point \(\boldsymbol{x}_p\) denotes the position of the root tip, see Eq. 9. The root tip heading \(\boldsymbol{h}_x\) is the direction of growth of the local axis after applying the rotation R(α, β), see Fig. 2. The scalar field \(s(\boldsymbol{x})\) contains nutrient concentration, water content, pressure head or temperature. The value \({(\boldsymbol{h}_x)}_3\) denotes the zcoordinate of the vector \(\boldsymbol{h}_x\). Minimising the zcomponent \((\boldsymbol{h}_x)_3\) yields a vector \(\boldsymbol{h}_x\) pointing preferably downwards (with a negative zcomponent) and therefore describing gravitropism. Plagiotropism is obtained by minimising the absolute value \((\boldsymbol{h}_x)_3\), which leads to larger values of \((\boldsymbol{h}_x)_1\) and \((\boldsymbol{h}_x)_2\) and therefore, horizontal growth. Chemotropism is described by maximising \(s(\boldsymbol{x}_p + dx \ \boldsymbol{h}_x)\); thus, we multiply by ( − 1) to obtain a minimisation problem. Objective functions can be freely combined (e.g.: by linear combination). In this way, different kinds of tropisms can be realised for each root type.
Root tip deflection
In many experiments like pot or rhizobox experiments, root growth is spatially bounded (Doussan et al. 2006). We can bound our root growth simulations by an arbitrary geometry which is given implicitly by a signed distance function. The signed distance function determines how close a given point is to a boundary and returns a negative value if the point is outside the boundary. Additionally, this provides a way to include obstacles in our model.
The following algorithm takes the spatial boundaries into account. In a first step, the rotation angles α and β are derived as described in Section “Tropisms”. If the new root tip position does not lie within the geometric boundaries, then a new pair (α, β) is chosen as follows: First, only β is chosen uniformly random between − π and π while α is left unchanged. If, after a maximal number of trials n _{ β }, no new valid pair α and β has been found, α is increased for a small fixed angle dα and the procedure for finding an angle β is started again. This simple approach leads to a realistic root behaviour at the boundaries, where thigmotropism can be observed.
Coupling to a soil model

A specific tropism can be chosen which depends on soil properties (e.g. water content, nutrient concentration or temperature). Depending on which plant and soil interaction model is described, combinations of different tropisms such as hydrotropism, chemotropism or thermotropism can be considered, see Eq. 11.

The root growth function λ(t) can be related to soil properties, which alters the roots’ elongation rate.

The branching strategy can depend on soil properties. In this way the density of lateral branches can vary or the root type of the branches can change.
The root growth model supplies the time dependent threedimensional geometry of the root system and resulting global parameters such as total length, surface or spatial distributions, see right part of Fig. 4. Generally it is computationally too expensive to directly use the explicit threedimensional geometry even for the static case. For this reason, density distributions are used to create a sink term in the plant and soil interaction model, which is often described by partial differential equations and numerically solved by the finite element method (Doussan et al. 2006; Javaux et al. 2008). The development of a suitable sink term is crucial and several upscaling techniques are available (Roose and Schnepf 2008). However, these methods often use simplifying assumptions, e.g. that roots are evenly distributed, that soil is homogeneous or that roots are functionally equivalent. Therefore, in general, sink terms need an accurate validation on an experimental basis.
We emphasise that the Matlab code for the root growth model is freely available.^{1} Every production rule has a Matlab file with a corresponding name. This allows to adapt existing production rules and to easily extend the model for new production rules.
The case of phosphate uptake
Results
We first present the results of the root growth model only, exemplified for maize. We compared our LSystem root growth model with the continuous model of Roose et al. (2001) where no stochastic processes are included. Then we simulate the interaction between root system and soil, exemplified for phosphate uptake. We investigated spatial properties of a maize root system confined in a pot and analysed the effect of chemotropism and gravitropism on phosphate uptake by coupling the root growth model to a soil model. Finally, sink terms with different complexities were derived from the simulated root architecture and their effect on phosphate uptake was analysed.
Root growth and architecture
Phosphate uptake from a pot
Having obtained the root architectural traits from our root system model, these results were integrated into a soil model to analyse phosphate uptake. The soil model is given by Eqs. 12–16. The root system grew in a pot with a bottom radius of 3 cm, a top radius of 5 cm and a height of 10 cm. Model parameters were taken from literature and represent typical values for phosphate uptake from soil by maize (Tinker and Nye 2000; Föhse et al. 1991). The initial concentration c _{0} was assumed to be 10^{ − 4} μmol cm^{ − 3}, impedance factor f = 0.3, water content θ = 0.4, buffer power b = 100, diffusion coefficient for phosphate \(D_l=10^{5}\) cm^{2} s^{ − 1}, maximal influx F _{ m } was assumed to be 2.76 ·10^{ − 7} μmol cm^{ − 2} s^{ − 1} and the Michaelis Menten constant \(K_m = 4 \cdot 10^{4}\) μmol cm^{ − 3}.
The phosphate uptake was dependent on the root surface area s per volume of soil. In every time step Δt of 1 day the values of s were determined from the root growth in volume elements of 0.125 cm^{3}. The overall simulation time was 50 days. We used the finite difference method to obtain a numerical solution of the solute transport model.
Discussion
LSystems are a common tool in plant architecture modelling (Prusinkiewicz and Lindenmayer 1990). They have also been used for root architecture modelling, although mainly in the context of visualisation (Prusinkiewicz 1998). We realised an interface in our LSystem model which enables coupling the threedimensional root growth model to any arbitrary soil model. Furthermore, the modular implementation in Matlab allows to define new production rules as required. Most dynamic root system models are based on simple production rules that include emergence of new main axes, growth of the axes and branching (Doussan et al. 2003). LSystem formalism is most suitable to describe such production rules. We explicitly stated our production rules and provided the corresponding Matlab files to increase reproducibility.
The way tropisms were implemented in our model differs from previous implementations. The most common approach is to compute the new growth direction by adding vectors denoting the initial growth direction, mechanical constraints and gravitropism (Pagès et al. 1989). Tsutsumi et al. (2003) compute the new growth direction in their twodimensional model based on differences of elongation rates at the elongation points (located opposite to each other just behind the root tip). In our implementation, the new direction was computed by random minimisation of an objective function. Tropisms were realised by choosing appropriate objective functions. The advantages of this approach are that tropisms can be easily described, see Eq. 11, and that results do not depend on the spatial resolution.
We used a maize root architecture to demonstrate basic features of the proposed model. Maize root lengths found in literature strongly vary depending on maize cultivar and environmental conditions. The presented results lie well in this range and do for example quantitatively correspond to the nitrate inefficient maize breed Wu312 described in Peng et al. (2010).
Root water and nutrient uptake based on threedimensional root architecture models is commonly included in soil models via sink terms (e.g. Clausnitzer and Hopmans 1994; Somma et al. 1998; Dunbabin et al. 2002, 2006; Doussan et al. 2006; Javaux et al. 2008). Those sink terms depend on root length or root surface densities. The spatial resolution of the soil model for which the densities are computed is often 1 cm^{3}. In our simulations we used half the edge length, i.e. the grid size was 0.125 cm^{3}. However, in our model, the spatial resolution can be freely chosen by the user depending on the scale of the problem. Ge et al. (2000) use the SimRoot model (Lynch et al. 1997) to simulate phosphate acquisition efficiency in dependence on gravitropism. Contrary to our model, this root architecture model is not coupled to a dynamic soil model. It uses the diffusion length of phosphate in soil, \(L=2\sqrt{D_e t}\), where D _{ e } is the effective diffusion coefficient, in order to estimate the depletion volume for a root system. No other mechanisms such as root responses to phosphate concentration are considered. This approach provides an estimate for the potential phosphate uptake under the assumption that diffusion is the dominant mechanism in the soil. Somma et al. (1998) base their sink term for nutrient uptake on the root surface density and the local averaged nutrient concentration. Depletion around individual roots is neglected. We used a similar approach in the ‘simple’ and ‘age dependent’ sink term but also provided an example where individual depletion zones were taken into account. In our example water transport was neglected. However, our LSystem model was designed in such a way that it can be coupled to models including water transport, e.g. based on the Richards equation as used by Somma et al. (1998). This would enable to study hydrotropism. Advances with regards to additional rhizosphere traits have been made by Dunbabin et al. (2006) by simulating the effects of phospholipid surfactants on water and nutrient uptake. In our example, we did not consider exudation. However, due to the flexibility of our approach, the root growth model can easily be coupled to soil models including exudation. The sink term for nutrient uptake in Dunbabin et al. (2002, 2006) is based on plant demand as well as nutrient concentration at the root surface according to the Baldwin, Nye and Tinker equation. Thus, depletion around individual roots is approximated with a steady rate solution. This is a similar approach like our sink term which considered individual depletion zones based on an approximate analytic solution to the dynamic model (Roose et al. 2001).
In this work we exemplified the coupling of the root system to a soil model with a model for phosphate uptake by a maize root system from a pot. We considered root response via chemotropism as described in Section “The case of phosphate uptake” and via root responses to barriers as described in Section “Root tip deflection”. We compared three different sink terms for nutrient uptake that were based on the root growth model. We showed that neglecting the development of depletion zones around individual roots is likely to overestimate uptake, in particular for nutrients with low mobility such as phosphate. The comparison of the different sink terms indicated that it may be difficult to distinguish between age and depletion effects experimentally in soil. Furthermore, we showed that chemotropism strongly increased phosphate uptake from a pot which is in good agreement with Jackson and Caldwell (1996). As demonstrated by Ge et al. (2000), this is due to overlapping depletion zones in the case of gravitropism.
A comparison to measurement results from literature on phosphorus uptake of maize in pot experiments required consideration of the soil volume, as well as the growing duration of the maize crop. We converted the results from literature using the pot volume of 509.5 cm^{3} as taken for our simulation. Data range from 28.74 to 51.66 μmol after 32 days (El Dessougi et al. 2003), and 33.90 to 83.31 μmol after 45 days (Mujeeb et al. 2008) phosphate root uptake depending on fertilization level and fertilizer type. These amounts are higher, but in a similar order of magnitude, as compared to our model outputs. A lower uptake is anticipated considering that we described a low phosphate scenario.
The main benefit of our approach in Matlab is that the LSystem model can be easily coupled to arbitrary soil models. Thereby various rhizosphere traits such as nutrient and water uptake and exudation can be modelled in response to root system development and vice versa. The possibility to let the root system grow in a confined environment will be useful for comparison to experimental data from pot or rhizotron experiments.
Conclusions
We presented a new dynamic root architecture model based on LSystems. The objective of this new model was to realistically represent a threedimensional root system that was subsequently coupled to a soil model in order to describe the dynamic interactions of both the root system with soil processes (water, solute transport) as well as local soil properties with the rooting pattern (root plasticity, tropism). Lsystems provide a compact description of dynamic root system models. Our implementation of the model into Matlab makes it accessible to a wider public and encourages its development with new features. Furthermore it facilitates coupling to different soil models.
We exemplified the proposed root growth model and its interaction with a soil model by simulating phosphate uptake of a maize plant in a confined growth environment (pot experiment). This example demonstrated that our approach provides a convenient tool to create feedback loops between root system and soil. Thus it is possible to analyse both root effects on soil processes via a sink term derived from the architecture model as well as soil property impacts on the root architecture dynamics via tropisms. We implemented chemotropism and were able to clearly reveal substantial effects of root architecture and its interaction with the soil environment on plant nutrient use. Chemotropism increased phosphate uptake by as much as 82% compared to a root system governed by gravitropism only. We compared three different sink terms and analysed their effect on calculated phosphate uptake and found that considering individual depletion zones around each root reduces calculated uptake significantly.
In contrast to existing root growth models, our model can be freely coupled to any soil model and new sink terms can be easily implemented as required. With this approach, we hope to facilitate model development and increase reproducibility of simulation studies.
Footnotes
 1.
The Matlab files can be downloaded at www.boku.ac.at/marhizo .
Notes
Acknowledgements
This work was supported by the Vienna Science and Technology Fund (WWTF, Grant No.: MA07008) and the Austrian Science Fund (FWF, Grand No.: T341N13).
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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