Applications
Fitting
Plant structure fitting - Branching
Retrieving branching laws
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Branching modelling and qualitative description is often a heavy task.
On each phytomer, several axillary branches may appear, with immediate or delayed development.
This diversity can also be found on a given axis, even within the same growth unit, from one phytomer to another, as defined in the botanical arrangements of branching (acrotony for instance).
However, in the GreenLab approach, the model focuses on the number of phytomers of a given type (physiological age) appearing at a given growth cycle, without worrying about their positioning.
For each type of axis (i.e. each physiological age), the law of branching can thus be assessed from the ratio of the number of branched phytomers to the total number of phytomers.
Hence, this single ratio can be defined as a branching probability pa. In practice, the definition of this ratio should be defined from a significant population, at a given rank, since this ratio may change along the axis.
Moreover, in some cases, the branching process is delayed. The retrieval of dormancy can be define from an awakening probability in the branching process, for each growth cycle.
Finally, the branching process may show couplings on a local scale, in a whorl or from one phytomer to its predecessor.
Simple coupling in branching.
On a given phytomer, let us consider the case of couplings in a simple whorl that may have no, one (right or left) or two branches (right and left) per whorl. The branching states can be described as follows:
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state (0,0): no branching on the phytomer
state (1,0): one branch, either Left, or Right
state (1,1): two branches on the phytomer, both Right and Left.
Let us now define n00, n10 and n11, as the respective number of the different states measured on a given population.
The total number of phytomers is therefore n = 10 + n10 + n11
The number of potential branches is 2.n since each phytomer may carry two branches.
The probability pa of branching can then be estimated by the single ratio
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pa = (n10 + 2.n11) / 2.n
( pa = 0.5 (n10 + 2. n11) / (n00 + n10+ n11) )
The thoeretical distribution among the three states can then be estimated without a coupling hypothesis .
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the state (0,0) number should appear close to n * (1 - pa) * (1 - pa)
the state (1,1) number should appear close to n * pa * pa
the state (1,0) number should appear close to 2 * n * (1 - pa) * pa
If significant differences appear, the two potential branching processes on a single phytomer must not be considered as independent.
A simple coupling model can be tested in such a context.
Introducing a coupling r to the branching model, we can write the probability of branching dependeing on the 3 states as follows:
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p(0,0) = (1-pa)(r+(1-r)(1-pa))
(coupling of no branching)
p(1,0) = 2(1-r) pa (1-pa) (no coupling)
p(1,1) = pa*(r + (1-r)*pa) (coupling of branching)
The coupling coefficient r can then be retrieved using the real distribution of a state.
Example. Coffee tree branches
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Phytomers on the coffee tree main stem usually have two branches.
Three states (0,0): no branch, (1,0): a single branch, (1,1): two branches are thus possible.
For a given population, measurements led to this distribution:
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Number of states (0,0) : n00 = 39
Number of states (1,0) : n10 = 15
Number of states (1,1) : n11 = 96
The number of branches is thus: n10 + 2 n11 = 15 + 96*2 = 207
And the number of phytomers is n = n00 + n10 + n11 = 39+15+96 = 150, thus giving 300 potential branches
The branching ratio (probability) is thus pa = 207/300; pa = 0.69
If no coupling is considered, the number of the different states should thus be:
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for state (0,0) : (1-pa)*(1-pa)*150 = 14 (14.415 to be compared with 39)
for state (0,1) : 2*a*(1-pa)*150 = 64 (64,17 to be compared to 15)
For state (1,1) : pa*pa*150 = 71 (71.415 to be compared with 96)
This distribution is quite different from the measured one, showing higher distributions for the (0,0) and (1,1) states and less single branching.
A phytomer thus shows a trtendency to develop either two branches or none.
The coupling model can be used in such a context.
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The theoreticall number of unbranched phytomers is:
p(0,0) = (1-pa)(r+(1-r)(1-pa))
By identification we have: n00 / n = 0.31* (0.69 r + 0.31)
giving
- r = 0.767