Team Digiplante

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Section: Scientific Foundations

Botanical Instantiations in GreenLab Model

At Metamer Level

Participant : X. Zhao (GreenLab associated team, Liama).

Figure 1. Dual scale Automaton for Plant Organogenesis.

In most cases, a dual scale automaton (DSA) is sufficient to describe the full organogenesis. The Automaton controls the bud mutation in different states named physiological ages. The number of physiological ages ((PA.) is small (less than 10). The Plant is organized in Metamers (microstates) and Growth Units (macrostates). Each metamer is a set of organs (internode, leaf, fruits and axillary buds). It is more efficient to create metamers than organs one by one because it gives directly the organ production and speeds up the computing of organogenesis and plant demand. Each growth unit is a set of metamers and the repetition of GUs gives birth to an axis so called ``Bearing Axis'' (BA).

At Substructure Level

The terminal bud with a given PA produces different kinds of metamers bearing axillary buds of various PA. These buds give birth to axillary branches. Even the PA of the main bud can change by mutation. This phenomenon is represented in the automaton as a transition between macro-states. These processes automatically create substructures. A substructure is characterized by its physiological age PA and its chronological age CA. All the substructures with the same PA and CA are identical if they have been set in place at the same moment in the tree architecture. Let us consider the example of a particular 100 year old tree. Its trunk is of PA 1, main branches of PA 2 and live about 15 years, twigs of PA 3, 4, 5 and respectively live about 7, 5, 2 years. Here, the total number of substructures with different PA and CA is about 30. It is small, even if the total number of organs is high. These substructures will be repeated a lot of times in the tree architecture, but they need to be computed only once for each kind of PA and CA. The tree production and construction will be obtained by stacking the substructures in the right way.

Factorization of Plant Development

Participant : H.P. Yan (GreenLab- Liama associated team to Digiplante).

In the case of parallel simulation, counting the number of organs is a typical bottleneck; the computing time can be tremendous for big trees and forests. To overcome this difficulty, GreenLab model takes advantages of the plant architecture organized thanks to the concept of PA and generated by the DSA.. Similar substructures (of same PA and CA) are found in the main architecture many times.

Figure 2. Plant factorization in substructure.

Suppose a tree with m PA and finite growth for the axes: the repetition of macrostates (i.e. the number of GU) of PA=k is equal to NK . Beyond this limit, the terminal bud can undergo a mutation and change PA (say k + 1 ), or die if k = m . So there are m kinds of substructures here that are represented by arrays whose fields contain the cumulated number of metamers according to their PA. A structure Skt is defined by its chronological age CA = t and its physiological age PA = k. It contains all the cumulated numbers of metamers produced from its birth until GC t.

Im1 ${\mfenced o=[ c=] S_1^t=\mfenced o=[ c=] s_{1,1},s_{1,2},\#8943 ,s_{1,m}^t,\mfenced o=[ c=] S_2^t=\m ...  0,s_{2,2},\#8943 ,s_{2,m}^t,\#8943 ,\mfenced o=[ c=] S_m^t=\mfenced o=[ c=] 0,0,\#8943 ,s_{m,m}^t.}$

All the items si, j with j<i are null because of the production rules. Structure S1t sums up all the metamers produced at GC t, for the whole plant. Let uk be the number of metamers per GU for a given PA k and ni, j be the number of substructures of PA j branched on the ith GU of the bearing axis of PA k. We have to stick the lateral and terminal substructures directly on the bearing axis of PA k, according to their positions as follows:

Im2 ${\mfenced o=[ c=] S_k^t=t·\mfenced o=[ c=] u_k+\#8721 _{i=1}^{t-1}\#8721 _{j=k+1}^m\mfenced o=( c=) n_{k,j}·\mfenced o=[ c=] S_j^i~\mfenced o=( c=) t\#8804 N_k{).}}$(1)

Ift>Nk , and along the trunk, an apical terminal substructure of physiological age k + 1 is born thank to the terminal bud mutation , so we have:

Im3 ${\mfenced o=[ c=] S_k^t=N_k\mfenced o=[ c=] u_k+\#8721 _{i-t-N_k}^{t-1}\#8721 _{j=k+1}^m\mfenced o=( c=) n_{k,j}·\mfenced o=[ c=] S_j^i+\mfenced o=[ c=] S_{k+1}^{t-N_k}~\mfenced o=( c=) t\#8804 N_k,t\lt m{).}}$(2)

This plant construction algorithm is very fast. Obviously, the computation time depends only on t*m and not on the number of organs produced. The substructures are constructed by a double loop, i.e., bottom up from the youngest CA=1 to the final CA=t and top down from the oldest PA=m to PA=1. A library of substructures is created for each PA and CA and will be used to build substructures of older CA and younger PA. As the number of organs per metamer is botanically known, GreenLab provides a mathematical tool that enables to compute the organ production of a virtual plant very quickly and thus suppresses the drawback of counting the number of organs one by one by simulation. This also leads to an efficient way to compute the plant demand that is no more than the scalar product between the number of organs and their corresponding sinks.

Computing the Biomass Production

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 immediately derived from formula (1 ) and (2 ) giving the number of metamers in the plant as we know the number of organs per metamer and their durations.

Biomass acquisition

Every leaf produces biomass that will fill the pool of reserves according to an empirical nonlinear function depending on its surface A, on parameters r1 , r2 , and on water use efficiency at GC k:E(k) . We suppose that the size of a leaf depends on its cycle of apparition (because of expansion). Let NKL be the number of leaves produced at GC k, known from Equation (1 ), the plant biomass production is:

Im4 ${Q_t=\#8721 _{k=1}^tN_k^L·f\mfenced o=( c=) A_k,r_1,r_2{,E(k)}.}$(3)

The empirical function chosen for the leaf functioning in GreenLab is:

Im5 ${f\mfenced o=( c=) A_k,r_1,r_2,E=\mfrac E{r_1/A_k+r_2}.}$(4)

This function can be easily changed according to modellers' choices.

For example the Light can be chosen as the driving force and we will use the Beer Law to compute the light interception by the leaves. Equation (4 ) is then replaced by:

Im6 ${Q_t=\mfrac E_tr\mfrac S_pk\mfenced o=( c=) 1-exp\mfenced o=( c=) -k\mfrac \mstyle {\#8721 _{j=1}^{n(t)}A_j}S_p}$(5)

where r is the resistance related to the transpiration of the leaf area Im7 ${(\#8721 A)}$ , k is the coefficient related to the light interception, Et the light use efficiency at cycle t and Sp a surface related to the crown projection.

Biomass partitioning

Each organ has a potential biomass attraction value that we name sink or organ demand. This sink pk(i) depends on the organ PA k and on its CA i (because of exapansion). The shape chosen for 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 CA, it must be flexible enough to give bell shapes, c or s shapes, etc.

We define the plant demand at GC n as the total biomass attraction of all organs (leaves, internodes, fruits, layers, roots, ...):

Im8 ${D_n=\#8721 _{o=L,I,F}\#8721 _{i=1}^tN_{t-i+1}^op_o{(i).}}$(6)

The Nko are given by Equation (1 ). It gives instantaneously the biomass $ \upper_delta$qi, to allocated to an organ of type o created at GC t-i + l and its total cumulated biomass qi, no :

Im9 ${\#916 q_{i,t}^o=\mfrac {p_o{(i)}}D_tQ_{t-1},~q_{i,t}^o=\#8721 _{j=i}^t\#916 q_{i,j}^o.}$(7)

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

As functions for organ sinks need to be flexible enough to capture the sink variation. Beta laws were found to be suitable for the purpose.


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