Deciphering the ontogeny of a sympodial tree

Abstract

This paper addresses the identification and characterization of developmental patterns in the whole structure of a sympodial species, the apple tree. Dedicated stochastic models (hidden variable-order Markov chains) were used to (i) categorise growth units (GUs) on the basis of their morphological characteristics (number of nodes and presence/absence of flowering) and position along axes, (ii) analyse dependencies between successive GUs and (iii) identify repeated patterns in GU sequences. Two successive phases, referred to as “adolescent” and “adult”, were identified in two apple tree cultivars. In the adolescent phase, “very” long monocyclic GUs were followed by long polycyclic GUs, whereas in the adult phase medium GUs were preferentially followed by short GUs. Flowering GUs constituted a preferential pathway between vegetative GUs of decreasing vigour (long, medium and short) and generated patterns that were interpreted with respect to fruiting regularity. The proposed modelling gave a global and quantitative picture of the two-scale structuring of apple tree ontogeny: a coarse scale corresponding to the succession of the previously mentioned phases and a fine scale corresponding to the alternation between flowering and vegetative GUs. This led us to propose a synthetic scheme of apple tree ontogeny that combines growth phases, polycyclism and flowering, and which could be transposed to other sympodial trees.

Appendices

Appendix A: Selection of the memories of a variable-order Markov chain

The order of a Markov chain can be estimated using the Bayesian information criterion (BIC). For each possible order r, the following quantity is computed

$$ BIC\left( r \right) = 2\log L_{r} - J^{r} \left( {J - 1} \right)\log n, $$

(1)

where L _r is the likelihood of the rth-order estimated Markov chain for the observed sequences, J ^r(J − 1) is the number of independent transition probabilities of a J-state rth-order Markov chain and n is the cumulated length of the observed sequences. The principle of this penalized likelihood criterion consists in making a trade-off between an adequate fitting of the model to the data [given by the first term in (1)] and a reasonable number of parameters to be estimated (control by the second term, the penalty term). In practice, it is infeasible to compute a BIC value for each possible variable-order Markov chain of maximum order r ≤ R since the number of hypothetical memory trees is very large. An initial maximal memory tree is thus built combining criteria relative to the maximum order and to the minimum count of memory occurrences in the observed sequences. This memory tree is then pruned, using a two-pass algorithm which is an adaptation of the Context-tree maximizing algorithm (Csiszár and Talata 2006): a first dynamic programming pass, starting from the terminal vertices and progressing towards the root vertex for computing the maximum BIC value attached to each sub-tree rooted in a given vertex, is followed by a second tracking pass starting from the root vertex and progressing towards the terminal vertices for building the memory tree.

Appendix B: Definition of parametric observation distributions for hidden variable-order Markov chains

In the case of a count variable such as the number of nodes of a GU, the observation distributions are parametric discrete distributions chosen from among binomial distributions, Poisson distributions and negative binomial distributions with an additional shift parameter d.

The binomial distribution with parameters d, n and p (q = 1 − p), B(d, n, p) where 0 ≤ p ≤ 1, is defined by

$$ b_{j} \left( y \right) = \left( {\begin{array}{*{20}c} {n - d} \\ {y - d} \\ \end{array} } \right)p^{y - d} q^{n - y} ,\quad \quad y = d,d + 1, \ldots ,n. $$

The Poisson distribution with parameters d and λ, P(d, λ), where λ is a real number (λ > 0), is defined by

$$ b_{j} \left( y \right) = \frac{{e^{ - \lambda } \lambda^{y - d} }}{{\left( {y - d} \right)!}},\quad \quad y = d,d + 1, \ldots $$

The negative binomial distribution with parameters d, r and p, NB(d, r, p), where r is a real number (r > 0) and 0 < p ≤ 1, is defined by

$$ b_{j} \left( y \right) = \left( {\begin{array}{*{20}c} {y - d + r - 1} \\ {r - 1} \\ \end{array} } \right)p^{r} q^{y - d} ,\quad \quad y = d,d + 1, \ldots $$