Roger Buis

The growth of the vein system of the leaf ofTropaeolum peltophorum was studied (i) at the vein level (analysis of the growth kinetics and of the growth field of the 7 primary veins) and (ii) at the limb level (comparison of the growth of the veins).The vein growth kinetics depend on the position of the vein on the leaf: there is a proximal/distal gradient and a bilateral gradient of the growth parameters (=variation of the temporal organization of the kinetics). The growth field of the veins (=local growth pattern) is also dependent on time and on vein position.

Heteroblastic development is often observed in Cormophytae, but it can also be characterized in Thallophytae as shown by the detailed investigation of the development of the algaAntithamnion plumula (Ceramiaceae, Rhodophyta). In this species, heteroblasty concerns (i) dimensional variables (such as pleuridia length and lateral cladome first tagma length) and (ii) variables that characterize the cell growth kinetics (main axis cells). Apex curvature also varies during ontogenesis.The generality of the property in plants led to search for its origin: apical meristem own dynamics, ontogenetic evolution of internal relationships, dialectical relationships between the plant and its environment.

L'analyse d'une cinétique de croissance y est conduite à partir du modèle logistique généralisé de Richards-Nelder. On distingue 2 types de processus dits mono- et multi-logistique. Dans le cas mono-logistique, le phénomène est correctement décrit par une seule fonction logistique. La cinétique de croissance est alors caractérisée par lea propriétés de chacune des phases G 1 à G 4, délimitées par les points singuliers Γmax, V max et Γmin. On appelle structure de croissance la contribution relative de ces différentes phases à l'expression de la croissance totale. Cette distribution temporelle de l'activité de croissance est une représentation discrétisée de la trajectoire y 0 → y max. On appelle processus multi-logistique toute cinétique de croissance correspondant à une somme de 2 ou plusieurs logistiques généralisées. II existe alors un plus grand nombre de points singuliers et donc un plus grand nombre de phases G i que dans le cas monologistique. La structure de croissance définit avec précision la cinetique du phénomène, ainsi que l'importance relative des différentes composantes logistiques au cours du temps. Une application en est donnée avec le grandissement des cellules pleuridiophores de l'Algue Antithamnion plumula. L'allongement de ces cellules est toujours de type bi-logistique. Les caractéristiques de la structure de croissance permettent une discrimination nette du sporophyte et des gametophytes mâle et femelle. Tandis que la composante 1 est pen différente, la composante 2 varie fortement selon la ploïdie du thalle. En revanche, il y a pen de différences entre les deux gamétophytes. Cette notion de structure de croissance est ainsi susceptible d'apporter des résultats originaux. Audelà de cet example, cette approche est proposée comme une méthode de caractérisation simple mais précise de toute cinétique de croissance basée sur lea variations temporelles de l'activité de croissance [couple ].

This communication presents a discussion of some extensions of the formalism of Verhulst's simple logistics, which may constitute an autonomous growth model of a more general scope.For that purpose, the basis concept of growth diagram or trajectory is called upon, as it affords the graphic representation of the change in the growth variable y, using two relevant kinetic parameters: the instantaneous rate and the instantaneous acceleration. The two possible kinds of trajectories are in relation to the use of absolute (V = dyldt; = dV/dt) or relative (or specific) values (R = (1/y)(dy/dt); R = dR/dt).

The growth of the vein system of the leaf ofTropaeolum peltophorum was studied at the vein level and at the limb level.The vein growth kinetics depend on the position of the vein on the leaf: there is a proximal/distal gradient and a bilateral gradient of the growth parameters. The growth field of the veins is also dependent on time and on vein position.Specific correspondences exist between singular points of the global growth curves, the variations of the growth field and the vein position. The spatio-temporal organization of the vein system growth is then characterized by positional variations of kinetic order which show the existence of areal morphogenetic dissymmetry.

The logistic function now constitutes the most widely used model for there presentation of growth kinetics of the continuous monotonous type in biological systems (populations, organisms, organs, ...). This ubiquity led to consider logistics from a phenomenological rather than mechanistic viewpoint. Whence the question : can logistics be given an interpretation, a signification which confers the rank of an "explicative" model to it? This Note presents some critical comments on the relationships between logistics and three types of biological systems : population demography, environmental resources, autocatalyzed reactions. The so-called functional (in the mathematical meaning) interpretation, which is then discussed, is based upon a variational principle : the occurrence of a minimum of a function associated with the logistic law. Its present limitation to the only simple logistics of Verhulst and the difficulties of its expression in biological terms are then pointed out.

The growth activity of an organ (variable y) is defined simultaneously by the instantaneous absolute ratedy/dt and its variationd 2y/dt2. The use of these two descriptors allows a sigmoidal (i.e. continuous and non periodical, as observed for the logistic function) growth curve to be discretized into a series of 5 growth states or phases which are delimited by the following singular values: max, Vmax (=0), max, adult stage. The (V, ) plot, termedgrowth trajectory, visualizes, e.g. in the case of Richards-Nelder's generalized logistics, the unequal duration of these phases and allows the dissymmetry property of the growth function to be characterized precisely.At a more integrated organization level, such anorganic series (e. g. the set of the leaves borne by a same branch), the above concepts allow two kinds of analyses to be carried out: (i) in thekinetic approach, the growth trajectory of the series can characterizes, under certain conditions, the emergence of a new property: theoccurrence of growth activity rhythms which are independent of meristem dormancy processes; (ii) thedemographic approach shows thegrowth structure of the phytomer population (a particular form of age structure) and the associated dimensional structure (plant partition into zones or regions in relation to the growth state of the various phytomers).

The filamentous and branched thallus of Antithamnion plumula is constitued of two different kinds of branches with apical growth: the cladomial axes with a continuous or indefinite growth, and the pleuridia with a limited growth. The size of the pleuridia depends on their position with respect to the lateral cladomial axes.The growth kinetics of 35 pleuridia were analysed using Nelder's generalized logistics. Each sigmoidal curve, which was divided into four growth stages from the instantaneous acceleration variations, was thus characterized by ten kinetic parameters: lengths at the time of the changes in growth stage, durations of the growth stages and maximum growth rate.