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Plant structure fitting - Rhythmic development

In the case of rhythmic growth, axis developments have to be considered as a dual scale process.
At whole axis level, built from serial growth units, and at growth unit level, built from serial phytomers.

In both cases, tools and methodologies (based on the Bernoulli process) introduced for continuous growth can be used, with, of course, differentiated parameterization relative to each level of organisation.

Fitting of growth unit development

The distribution of the number of phytomers in a growth unit always follows a unimodal or a bimodal curve shape.

    The unimodal case (full pre-formation or full neo-formation)

    The unimodal case is related to a single functioning process, either pre-formed or neo-formed.
      The difference lies in the fact that, in the pre-formed case, the phytomer expands building the full growth unit, while in the neo-formed case, phytomers appear one after the other until a stop occurs related to a season, or to meristem death, or flowering (sympodial growth).
      These unimodal shaped distributions can be fitted to binomial laws.

      Growth unit construction is seen as a Bernoulli process with a probability b of each phytomer appearing, applied to a given number of growth cycles N.

      Fitting N and b parameters can be performed using measurements taken on a significant number of growth units, analysing their distribution according to their respective number of phytomers.
      This fit can then be simply deduced from the distribution expected value and variance, as defined for continuous growth.

    The bimodal case (pre-formation followed by neo-formation)

    The bimodal case is related to a mixed pre-formed neo-formed growth unit building process.
      A first section of the growth unit is built from pre-formed phytomers, followed by neo-formed phytomers.

      This situation is common in many temperate species (poplar tree, wild cherry tree, etc.) but also on tropical species (e.g. the cacao tree).

      For a given physiological age, such bimodal distribution shapes can thus be fitted with two binomial laws and the probability pn of having neo-formation.
      pn can be simply assessed by a ratio standing for the proportion of neo-formation occurrences.

      However, the neo-formed section may show (-or not) significant dispersion for the number of cycles.
      This neo-formed section can thus be adjusted to a binomial or to a negative binomial law.

      Such fittings can be solved using, for instance the least square approach, requiring implementation of the development model (and will not be detailed here).

      It is interesting to note that, on a single individual, the neo-formed part decreases with the physiological age. The neo-formed part can be systematically seen on the trunk, more or less established on a secondary axis, and not expressed at all on short axes.
      On some species, binomial laws parameters are fairly stable depending on the physiological age, and the various distributions can be fitted by the variation in the neo-formed section.