Project-Digiplante:The Equations of Plant Growth and Development

Inria / Raweb 2009
Presentation of the Project Digiplante

Section: Scientific Foundations

The Equations of Plant Growth and Development

GreenLab combines both organogenesis and functional growth in a mathematical dynamic system. A dual-scal automaton was initially used [78] to describe GreenLab organogenesis. However, it was shown to be equivalent to a growth grammar [4] , and this formalism is used from now on to describe the GreenLab organogenesis model. In this section we recall the basic botanical hypotheses used to construct the model of organogenesis, how it defines a growth cycle that is used to synchronize the computation of both organogenesis and functional growth. The construction of the growth grammar is also recalled, with its fundamental property: its factorization, taking advantage of all instantiations in plant architecture. Based on this growth grammar, the functional equations of biomass acquisition and distribution can be derived.

Plant Development and Growth Cycle

As explained in [52] , organogenesis results from the functioning of undifferentiated cells constituting the apical meristem and located at the tip of axes. When in active phase, this meristem forms buds that will develop into new growth units composed of one or several metamers (also called phytomers). A metamer is a botanical entity chosen as the elementary scale to model plant architectural development in this study. It is composed of an internode bearing organs: axillary buds, leaves, flowers. Depending on species, metamers are set in place rhythmically or continuously.

In the rhythmic case, see Figure 1 , the plant grows by successive shoots of several metamers produced by buds. The appearance of these shoots defines the architectural Growth Cycle . A Growth Unit is the set of metamers built by a bud during a growth cycle. These metamers can be of different kinds and ordered according to botanical rules, like acrotony. For example, most temperate trees grow rhythmically, new shoots appearing at spring. For such plants, and if we do not consider polycyclism and neoformation, the architectural growth cycle corresponds to one year.

Plant growth is said continuous when meristems keep on functioning and generate metamers one by one, see Figure 1 . The number of metamers on a given axis (that is to say generated by the same meristem) is generally proportional to the sum of daily temperatures received by the plant, see [59] . The growth cycle is defined as the thermal time unit necessary for a meristem to build a new metamer, it can be quite short, corresponding to a couple of days. The growth unit is thus simply composed of one metamer. The growth of tropical trees, bushes or agronomic plants is often continuous.

So far, Digiplante does not consider time scales that are smaller than the architectural growth cycle and we study the development of new growth units as a discrete process. The Chronological Age (CA) of a plant (or of an organ) is defined as the number of growth cycles it has existed for.

Figure 1. Growth Cycle (GC), Growth Unit (GU) and Metamer in GreenLab. The axis length is given as a function of time for continuous and rhythmic growths

Since metamers may bear axillary buds, plant architecture develops into a hierarchical branching system. [52] underlined that architectural units can be grouped into categories characterized by a particular combination of morphological parameters. Thus, the concept of Physiological Age (PA) was introduced to represent the different types of growth units and axes. For instance, on coffee trees, there are two types: orthotropic trunk and plagiotropic branches. The main trunk's physiological age is equal to 1 and the oldest physiological age denoted by P corresponds to the ultimate state of differentiation for an axis, it is usually short, without branches. We need less than 5 physiological ages to describe the axis typology of most trees. The apical meristem or bud of an axis is thus characterized by the physiological age of the growth unit that it may produce and a metamer is characterized by its physiological age i (which is the physiological age of the growth unit that it belongs to) and that of the buds that it bears j . Except in some very rare cases, we always have: i $\ge$ j . Moreover, along an axis, the morphological features of the growth unit may evolve with the age of the apical meristem. This process is described as the meristem sequence of differentiation by [52] , and corresponds to a transition to a superior physiological age of the meristem.

Alphabet, Growth Grammar and Structural Factorization

This section summarize results presented in [4] ,[2] ,[12] . In GreenLab, the alphabet $Im1 $\#119970 $$ is given by the set of metamers $Im2 $\#8499 $$ and buds $Im3 $\#8492 $$ . A metamer is defined with four indices and is denoted by m_pq^t(n) :

its chronological age: n ,
its physiological age: p ,
the physiological age of its axillary buds: q , (q $\ge$ p) ,
the chronological age of the plant: t .

A bud is defined by three indices and is denoted by b_p^t(n) :

its physiological age: p ,
the number of growth cycles k for which bud's physiological age has been p - in the sequel, we will call it ontogenic age of a bud,
the chronological age of the plant: t .

If T is the maximum growth time, the organogenesis alphabet is given by: $Im4 $\mstyle {\#119970 =\#8499 \#8746 \#8492 }$$ , with

$Im5 $\mfenced o={ \mtable{...}$$

(1)

We do not consider symbols for organs since the constitution of a metamer is supposed fixed by botanical rules (an internode and a given number of leaves and fruits). If, for example, flowering is particularly studied, symbols denoting flowers could be introduced in the alphabet.

Definition 1 (Set of words over an alphabet) The set of words over an alphabet $Im6 $\#119964 $$ is defined as the monoid generated for the concatenation operator ".", seen as an internal, non-commutative operation, by $Im7 ${\#119964 \#8746 {1}}$$ , where 1 is the neutral element for the concatenation operator (which corresponds to the empty word). It is denoted $Im8 $\#119964 ^*$$ .

$Im9 ${\#119970 }^*$$ will thus represent all the possible topological structures composed with buds and metamers. For example, $Im10 ${m_12^t{(1)}b_{2}^t{(0)}b_{1}^t{(1)}\#8712 \#119970 ^*}$$ represents at growth cycle t a structure composed of an internode of physiological age 1, bearing a lateral bud of physiological age 2 and an apical bud of physiological age 1. We will see that of course the structures of interest, that is to say botanically relevant, form a small subset of $Im9 ${\#119970 }^*$$ (corresponding to a language over $Im1 $\#119970 $$ ). It is important to consider concatenation as non-commutative when studying plant topology.

As recalled in section 3.1 , GreenLab organogenesis can be seen as the combination of two phenomena, branching and meristem differentiation. It can be easily modelled with the grammar formalism and we propose to define it as a F0L-system as follows:

Definition 2 (GreenLab Organogenesis) GreenLab organogenesis is defined as a F0L-system [65] [71] $Im11 $\mfenced o=〈 c=〉 \#119970 ,\#8492 ,P_r$$ with the following production rules P_r :

For all $Im12 ${{(t,n,p)}\#8712 ~{[0;T]}×{[0;min\mfenced o=( c=) \#964 (p),t]}×{[1;P]}}$$ :

$Im13 $\mtable{...}$$

and for all (t, n, p, q) $\in$ [1;T]×[1;t]×[1;P]×[p;P] :

$Im14 $\mtable{...}$$

with:

u_pq(t) : number of phytomers m_pq in a growth unit of PA p , appearing at growth cycle t
v_pq(t) : number of active axillary buds of PA q in a growth unit of PA p , appearing at growth cycle t
$\tau$ (p) : number of GC after which a bud of PA p changes to PA $\mu$ (p) . Vectors $\tau$ and $\mu$ characterize meristem differentiation.

A fundamental result [73] was surprisingly never used in the context of models of plant development: it is the ability to factorize the L-system productions. It showed particularly adapted to the concept of physiological age, cf. [6] , [2] . We are thus able to factorize plant structure into smaller parts that may repeat themselves a large number of times.

Definition 3 (Substructure) At growth cycle t $\ge$ 0 , a substructure of physiological age p, 1 $\le$ p $\le$ P and chronological age n, 0 $\le$ n $\le$ t is a word in $Im15 $\#119970 ^*$$ defined as the complete plant structure that is generated after n cycles by a bud of physiological age p . It is also characterized by the ontogenic age k of the bud generating it. It is denoted by S_p^t(n, k) (corresponding to the structure generated by b_p^t-n(k) after n growth cycles).

Figure 2. Substructures of physiological ages 1, 2, 3 at chronological ages 0, 1, 2 and their organization: S₁(0) , S₂(0) , S₃(0) are buds of physiological age 1, 2, 3 respectively. In this example, a growth unit of physiological age 1 is composed of 2 metamers of type m₁₃ and 1 metamer of type m₁₂ ; a growth unit of physiological age 2 is composed of 2 metamers of type m₂₃ ; a growth unit of physiological age 3 is composed of 1 metamer of type m₃₃ (without axillary bud).

We show how the structural factorization allows the inductive computation of all the substructures, providing an efficient algorithm to simulate plant organogenesis. We deduce the following important result.

Theorem 1 (Dynamic Equation of Plant Development) For all t $\ge$ 1 , $Im16 ${n\#8712 \mfenced o=[ c=] 1,t}$$ , $Im17 ${p\#8712 \mfenced o=[ c=] 1,P}$$ , $Im18 ${q\#8712 \mfenced o=[ c=] p,P}$$ , $Im19 ${k\#8712 \mfenced o=[ c=] 0,min\mfenced o=( c=) \#964 (p),t}$$ , we have:

$Im20 $\mtable{...}$$

(2)

This decomposition is illustrated on S₁(2) in Figure 2 . If we suppose that all the elements of the alphabet (set of metamers m_pq^t(n) , set of buds b_p^t(n) ) as well as the sequences $Im21 $\mfenced o=( c=) u_{pq}{(t)}_t$$ and $Im22 $\mfenced o=( c=) v_{pq}{(t)}_t$$ are known, Theorem 1 shows us how to build the topological structure of the plant at any growth cycle t recursively, as follows:

$\bullet$ Substructures of chronological age 0 are buds: $Im23 $\mstyle {S_p^t{(0,k)}=b_p^t{(k)}}$$ , $\bullet$ and if all the substructures of chronological age n-1 are built, we deduce the substructures of chronological age n from Equation 2 as functions of m_pq^t(n) , u_pq(t-n + 1) and v_pq(t-n + 1) .

Substructures and metamers will be repeated a lot of times in the tree architecture, but they need to be computed only once for each kind.

If we only consider topology, we do not need to characterize the metamers by their chronological ages nor by the plant age. m_pq^t(n) can thus be simply be denoted by m_pq . But if we consider the functional growth of a plant, metamers of different chronological ages have different masses and sizes, as well as metamers of the same chronological age, but at different plant ages.

[4] introduced how the dynamic development equation can be extended to build plant geometry with geometric operators replacing the concatenation operator. We will not detail this point here since our objective is to study plant functional growth which only relies on topology and not geometry in GreenLab.

Plant toplogy can simply be seen as a function of:

the sequences $Im21 $\mfenced o=( c=) u_{pq}{(t)}_t$$ and $Im22 $\mfenced o=( c=) v_{pq}{(t)}_t$$ for all (p, q) such that 1 $\le$ p $\le$ q $\le$ P (they will be called development sequences ),
the vectors $Im24 $\mfenced o=( c=) \#964 (p)_{1\#8804 p\#8804 P}$$ , $Im25 $\mfenced o=( c=) \#956 (p)_{1\#8804 p\#8804 P}$$ in case of meristem differentation.

The GreenLab organogenesis model has been derived in 3 forms:

GL1 corresponds to the deterministic organogenesis model, without influence of the plant functioning. Mathematically, it corresponds to u_pq and v_pq constant, see [18] . In such case, S_p^t(n) are (topologically) independent of t , and the construction does not have to be done at each growth cycle but only once.
GL2 corresponds to a stochastic model of organogenesis, u_pq and v_pq are stochastic variables. As a consequence, substructures of the same chronological and physiological ages can be very different: S_p^t(n) is a stochastic variable with values in $Im26 ${\#119982 }^*$$ . More details are given in section [8] and in [12] .
GL3 corresponds to a deterministic model with total retroaction between organogenesis and photosynthesis. u_pq(t) and v_pq(t) are functions of the biomass produced by the plant at growth cycle t-1 and t respectively, see [15] .

U^t will denote the vector $Im27 $\mfenced o=( c=) u_{pq}{(t)}_{1\#8804 p\#8804 q\#8804 P}$$ and $Im28 $\mfenced o=( c=) U^t_t$$ the associated sequence of vectors. Likewise, V^t will denote the vector $Im29 $\mfenced o=( c=) v_{pq}{(t)}_{1\#8804 p\#8804 q\#8804 P}$$ and $Im30 $\mfenced o=( c=) V^t_t$$ the associated vector of sequences.

Functional Growth

Literature is already abundant on the functional concepts underlying the GreeLab model and its various versions ([6] , [18] , [7] , [5] , [3] ). The fundamental principles common to all these versions are actually very general and are shared (to some extent) with various other models (for example LIGNUM [70] , TOMSIM [57] , GRAAL [55] ...). The central equation of GreenLab describes the growth of an individual plant potentially in a population (field crops, forest stands...). Competition with other individuals for light, water, nutrients can thus be taken into account and affects the computation of biomass production, see [3] regarding competition for light or [9] for competition for water.

GreenLab aims at describing the source-sink dynamics during plant growth. Sources correspond initially to the seed and then to biomass production and reserve remobilization. Sinks are demands for biomass of all living organs. So far, all sinks have access to all sources since we consider a common pool of biomass. However, this hypothesis is not fundamental and could be relaxed. Since structural development is described in GreenLab at the level of organs, the computation of demands is coupled with organogenesis. For this reason, a consistent time unit for architectural growth and photosynthetic production is defined in order to handle a constant structure. It allows the derivation of the discrete dynamic system of growth, see Figure 3

Figure 3. Flowchart of the GreenLab model.

It is not necessary to build the tree structure to compute biomass production and partitioning at a given chronological age. We only have to compute organ production, plant demand and photosynthesis. All these data can be derived recursively by applying structural morphisms [2] to the dynamic equation of plant development (2 ) giving the number of metamers in the plant as we know the number of organs per metamer and their life spans. The functional growth is then described by the two following steps, biomass acquisition (production) and distribution among living organs.

Biomass acquisition :

Different strategies were used and are currently used in GreenLab according to plants to compute Q^t , the biomass production at growth cycle t . There are mostly 3 sources to provide biomass for allocation: seed, resource acquisition by photosynthesis and reserve remobilization (from roots, layers or senescent organs). We will not speak about the seed, which usually gives u⁰ and may be considered as emptying on several growth cycles, that is to say also contributing to u¹ , u² ...

Classically, the biomass production at growth cycle t Q^t is taken as a parametric function of $Im31 ${\#119964 }^t$$ , the total photosynthetic leaf area, and of the environmental conditions. To keep it simple, we sum up all the environmental effects in E^t , which may be chosen as a function of the Photosynthetically Active Radiation (PAR), hydric stress (Fraction of Transpirable Soil Water), temperature (for biologic efficiency)... Only simple functions have been tested so far in GreenLab. A possible production function inspired by classical crop models and Beer-Lambert's law ([7] and [3] for details) can be written:

$Im32 ${Q^t=E^t\#946 \mfenced o=( c=) 1-exp\mfenced o=( c=) -\#947 {\#119964 }^t}$$ (3)

where $\beta$ and $\gamma$ are empirical parameters.

It is helpful to rewrite this equation by changing the parametric structure:

$Im33 ${Q^t=E^t\#956 S_p\mfenced o=( c=) 1-exp\mfenced o=( c=) -k\mfrac {\#119964 }^tS_p}$$ (4)

with the parameters having now a more relevant physical meaning: $\mu$ is an energetic conversion efficiency, k is the extinction coefficient of the Beer-Lambert law and S_p is related to a characteristic surface for resource acquisition.

In her PhD, V. Letort [64] proposed that, after some time, a proportion of organ biomass is given back to the common pool and reallocated to new organs in expansion.

Biomass partitioning :

We recall that m_p^t(n) denotes at growth cycle t a metamer of physiological age p and chronological age n . It contains organs of type o (where o = b , p , i , f , for blades, petioles, internodes, flowers or fruits respectively) whose masses are denoted by q_{o, p}^t(n) at growth cycle t . Let T_{o, p} denote the maximal life span of organs of type o and physiological age p . The allocation equation is thus given for all t $\ge$ 0 by:

$Im34 $\mfenced o={ \mtable{...}$$ (5)

where:

p_{o, p}^t(n) is the sink of an organ of type o in m_p^t(n)
D^t is the total demand of the plant at growth cycle t (that is to say the sum of all sinks)

$Im35 ${D^t=\munder \#8721 {o,p}\munderover \#8721 {n=0}T_{o,p}N_{o,p}^t{(n)}p_{o,p}^t{(n)}}$$ (6)
where N_{o, p}^t(n) is the number of organs of type o , physiological age p , chronological age n at time t .
Q^t is the biomass available for allocation.

Eventually, the organ volume depends on its apparent density and its dimensions on allometric rules. All this features can be measured directly from organ forms.

Remark 1 This general formulation concerns primary growth of the above ground organs. For the complete model, we need to consider the root system demand, and for trees, the ring demands (for the secondary growth). A detailed presentation of the different types of modelling strategies can be found in the PhD thesis of V. Letort [64] .

The shape chosen for the sink variation function p is up to the user, but it should be able to fit properly any kind of numerical variations of the sinks according to the organ chronological age. Beta laws were found to be suitable for the purpose.

Interactions between Plant Architectural Development and Functional Growth
During her PhD, A.Mathieu [67] modelled a retroaction of plant functioning on plant topology controlled by the state of trophic competition (GL3 model). Such model illustrates in a very simple way how plant plasticity is expressed under environmental constraints.
The vectors U^t and V^t that give the rules of the organogenesis model (that is to say that control the bud behavior) are chosen as functions of the ratio of available biomass to demand. Integer part of linear functions are usually chosen. In Figure 3.2.4 , we see the difference between the GL1 model (with constant sequences $Im28 $\mfenced o=( c=) U^t_t$$ , $Im30 $\mfenced o=( c=) V^t_t$$ ) and the GL3 model, (with sequences $Im28 $\mfenced o=( c=) U^t_t$$ , $Im30 $\mfenced o=( c=) V^t_t$$ functions of u^t ).
Figure 4. Effects of climatic variations on the GL1 and GL3 organogenesis models: on the left side (GL1), topology is fixed, organs are simply smaller to adjust to the environment; on the right side (GL3), toplogy is variable and the plant adapts its development to the environmental conditions.
U^t is a function of $Im36 $\mfrac Q^{t-1}D^{t-1}$$ and V^t a function of $Im37 $\mfrac Q^tD^t$$ (they are usually chosen as linear functions). The behaviour of the system was studied by A. Mathieu during her PhD. The conditions of the growth stabilisation were determined according to the parameters. Moreover, an interesting emerging property was obtained: under some conditions, rhythmic phenomena may be generated by the system (Biomass production, fruiting and branching alternation, number of internodes per growth unit ..., see Fig. 3.2.4 ). Such phenomena are often observed during plant growth but were poorly modelled so far.
Figure 5. Rythmic growth for fruiting and branching in alternation depending of the retroaction between plant production and plant development.
As detailed in the section giving the new research results of the team, some important results concerning the calibration of the thresholds of fructification have been obtained on cucumber plants (in collaboration with the Chinese Agricultural University and Wageningen).
The objective is now to give a parobabilistic framework to this model of interaction.

Extension of GreenLab to Field and Stand Levels
The results on single plant growth modelling are extended at the field and stand level, in order to simulate crop and forest production. It needs to integrate the competition for light and for soil resources among plants.
A model of competition for light :
The empirical production equation of GREENLAB is extrapolated to stands by computing the exposed photosynthetic foliage area of each plant. The computation is based on the combination of Poisson models of leaf distribution for all the neighbouring plants whose crown projection surfaces overlap, [3] .
To study the effects of density on architectural development, we link the proposed competition model to the model of interaction between functional growth and structural development introduced by Mathieu [15] .
Figure 6. Simulation of tree growth in heterogeneous conditions: view from above and detailed architectures of the individuals. In the upper right corner of the figure, Tree 1 grows in open-field like conditions. In the lower left corner, Tree 3 surrounded by its four neighbours (including Tree 2) severely suffers from competition
The model was applied to mono-specific field crops and forest stands.
The application of the model to trees illustrates the expression of plant plasticity in response to competition for light, cf. Figure 3.2.5 . Density strongly impacts tree architectural development through interactions with the source-sink balances during growth. The effects of density on tree height and radial growth that are commonly observed in real stands appear as emerging properties of the model.
For high density crops at full cover, the model is shown to be equivalent to the classical equation of field crop production (Howell and Musick, 1984) [58] . However, our method is more accurate at the early stages of growth (before cover) or in the case of intermediate densities. It may potentially account for local effects, such as uneven spacing, variation in the time of plant emergence or variation in seed biomass.
Functional landscapes :
Models of landscape functioning aims at simulating, crop plantations and small landscape with a “reactive” environment. The goal is to simulate water exchanges (rain, runoff on terrain and absorption, diffusion in soil, plant water uptake and evapotranspiration) and competition in interaction with the GreenLab growth model. The difficulty of the approach lies in the multiphysics and multiscal models to implement. After some prelimnary works carried out at LIAMA, it is now the PhD subjet of V. Le Chevalier, under the supervision of Marc Jaeger.
Two successive prototypes were developped. The first prototype, voxel based, was a simple simulator synchronizing all events at a daily schedule (water rain, run-of, diffusion, plant growth). Models were basic, and run-of simulated as a diffusion process on the land surface [62] , using a discrete volume spatial grid. This model was implemented in C++ under QT environment and tested on BULL Novascale HPC (16 cores). The second prototype, is still based on a spatial discrete grid, but limited to the layer. It involves an appropriate water run-off model, and the plant model is a simple version of the GreenLab crop model and involves more advanced visualisation tools.
At this stage, visualisation of functional landscape simulations aims at visualizing combination of maps (among terrain altitudes, water soil content, run off, daily biomass, cumulated biomass, temperature, ...). Classical surface mesh tools were written, as well as histogram, and curve display tools, allowing comparisons during a given period, or spatial heterogeneity comparisons at a given stage. The orginality of the developped tools lays in the fact that all these maps are dynamic (daily change), and thus not compatible with classical approaches (all in memory). Vegetation representation is rougth, limited to a simple color definition merged with the relief texture. A "pseudo realistic" landscape visualisation can be preformed representing the cumulated biomass by spheres, converting some how the crop production to a "tree crow". The correponding tool (SURFVIEW) is written in C++ with GLUT and OpenGL librairies.
The system was tested on synthetic cases, with real climate conditions and published in JCST journal [9] .
Both prototypes show strong conceptual limitations especially on two crucial points: model synchronism and data integrity. Time synchronism is a strong critical aspect in our case since we have to face calendar asynchronism processes (rain, run off) with thermal time cycles (plant development). However plant functioning must be simulated at short periods of calendar time in order to uptake the appropriate water supply. Concerning data now, and more precisely water ressources, the various models (run off, absorbtion, plant growth interact spatially on the same data with high risk of collusion). Since end 2006, concepts to develop the design of landscape functional simulators are extensivily studied leading slowly to discrete event simulation formalisms.

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