Roger Buis

On the generalization of the logistic law of growth

This communication presents a discussion of some extensions of the formalism of Verhulsts 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; Γ = dV/dt) or relative (or specific) values (R = (1/y)(dy/dt); ΓR = dR/dt).In the case of simple logistics, the trajectory (V, Γ) allows 4 growth phases or states to be distinguished. The diagram (R, ΓR) shows that the deceleration of the specific rate is not monotonous.In the case of Richards - Nelders generalized logistics, the qualitative variation of the growth trajectory depends on the value of the dissymmetry parameter (occurrence of a critical value which determines the number of growth states).Blumbergs model is characterized by an analogous property and, moreover, can account for a non monotonous variation of the specific growth rate.

Growth activity and structure at various organization levels in plants

The growth activity of an organ (variable y) is defined simultaneously by the instantaneous absolute ratedy/dt and its variationd2y/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: Гmax, Vmax (Γ=0), Гmax, adult stage. The (V, Γ) plot, termedgrowth trajectory, visualizes, e.g. in the case of Richards-Nelders 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).

Le phenomene d'heteroblastie chez les vegetaux: Comment l'expliquer?

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.Some models were put forward to accout for the development ofAntithamnion but did not take into account the heteroblasty phenomenon. The investigations presented here are a first attempt at representation through formal grammar, based on double wall map systems. The model leads to the assumption of the occurrence of an oscillatory mechanism associated with a threshold that determines both qualitatively (axial cell of the cladomophoric or pleuridiophic type) and quantitatively the development of the whorls.

Kinetics of pleuridial growth in Antithamnion plumula (Rhodophyceae)

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 Nelders 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.The data showed that all curves displayed a marked left dissymmetry which reflected the brevity of the accelerated growth stage (1/5 of total growth duration). The maximum acceleration stage occurs between the individualization of the pleuridial initial cell and the first mitosis. Previously known experimental data were related to this quantitative result so as to put forward a biological meaning. New research pathways were suggested.The kinetic parameters investigated were used for a classification of pleuridia. The smallest ones displayed the slowest growth: they needed twice more time to reach their maximum size than the largest pleuridia whose growth was the fastest. This association of traits was related to the occurrence of acropetal fluxes along the main axis of the thallus.

Leaf dissymmetry and vein growth field

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.Specific correspondences exist between singular points of the global growth curves (Vmax, Гmin), 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.

Kinetics of apical phyllogenesis of the main stem of soybean (Glycine max (L.) Merr.): Effects of stand density

Abstract The activity of the apical meristem of the main stem axis of soybean ( Glycine max (L.) Merr.), cv. Weber, is markedly dependent on stand density, which tends to decrease the number of leaves generated by the main axis apex, without significantly modifying the duration of development. The dependence of the organogenic activity of the apical meristem on stand density was analysed using Nelder s generalized logistics ( Biometrics 17 , 89–110, 1961). With increase in density, maximum phyllogenesis rates, which are observed earlier during the ontogenesis, are decreased. There is a decrease in the duration of the accelerated growth phase and an increase in the length of the decelerated growth phase. The number of leaves formed is directly related to the duration of these two phases.

Activation-inhibition model for branching of a filamentous organism

Abstract The branching pattern of the moss protonema is described by a dynamic model based on an activation-inhibition system with intercellular diffusion of an inhibitor. In this model, each cell can adopt one of two stable states, one quiescent and the other leading to branch initiation. The future of each cell depends on the state of the neighboring cells, as well as on its own initial state. This state is decided by a monotonic gradient of morphogen in the mother cell. This model thus explains certain principal characteristics of branching of the moss protenema with the aid of two notions: the predetermination of cells (memory of the mother-cell activity) and activator-inhibitor coupling. The possibilities of using this kind of model to describe the growth of other branched plants are discussed.

Cellular Growth and Cellular Division in Relation to Age and Illumination in the Protonema of Ceratodon purpureus BRID.

Summary The length of every primary cell (= cell issued from the division of an apical cell) from young protonema of Ceratodon purpureus is closely related to instantaneous speeds of elongation and mitosis of the filament. Under constant conditions (continued light or darkness) these different parameters present non periodic aleatory fluctuations. On the contrary, photo periodic conditions (12 h) induce a circadian rhythm of these characteristics.