One of the most intriguing questions in life science is how living organisms develop and maintain their predominant form and shape via the cascade of the processes of differentiation starting from the single cell. Mathematical modeling of these developmental processes could be a very important tool to properly describe the complex processes of evolution and geometry of morphogenesis in time and space. Here, we summarize the most important biological knowledge on plant development, exploring the different layers of investigation in developmental processes such as plant morphology, genetics, plant physiology, molecular biology and epigenetics. As knowledge on the fundamentals of plant embryogenesis, growth and development is constantly improving, we gather here the latest data on genetic, molecular and hormonal regulation of plant development together with the basic background knowledge. Special emphasis is placed on the regulation of cell cycle progression, on the role of the signal molecules phytohormones in plant development and on the details of plant meristems (loci containing plant stem cells) function. We also explore several proposed biological models regarding regulating plant development. The information presented here could be used as a basis for mathematical modeling and computer simulation of developmental processes in plants.

Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).

Plants and animals have highly ordered structure both in time and in space, and one of the main questions of modern developmental biology is the transformation of genetic information into the regular structure of organism. Any multicellular plant begins its development from the universal unicellular state and acquire own species-specific structure in the course of cell divisions, cell growth and death, according to own developmental program. However the cellular mechanisms of plant development are still unknown. The aim of this work was to elaborate and verify the formalistic approach, which would allow to describe and analyze the large data of cellular architecture obtained from the real plants and to reveal the cellular mechanisms of their morphogenesis. Two multicellular embryos of Calla palustris L. (Araceae) was used as a model for the verification of our approach. The cellular architecture of the embryos was reconstructed from the stack of optical and serial sections in three dimensions and described as graphs of genealogy and space adjacency of cells. In result of the comparative analysis of these graphs, a set of regular cell types and highly conservative pattern of cell divisions during five cell generations were found. This mechanism of cellular development of the embryos could be considered as a developmental program, set of rules or grammars applied to the zygote. Also during the comparative analysis the finite plasticity in cell adjacency was described. The structural equivalence and the same morphogenetic potencies of some cells of the embryos were considered as the space-temporal symmetries. The symmetries were represented as a set of regular cell type permutations in the program of development of the embryo cellular architecture. Two groups of cell type permutations were revealed, each was composed of two elements and could be interpreted as the mirror and rotational space symmetries. The results obtained as well as the developed approach can be used in plant tissue modelling based on the real, large and complex structural data.

Plant growth occurs due to cell proliferation in the meristem. We model the case ofapical meristem specific for branch growth and the case of basal meristem specific forbulbous plants and grass. In the case of apical growth, our model allows us to describethe variety of plant forms and lifetimes, endogenous rhythms and apical domination. In thecase of basal growth, the spatial structure, which corresponds to the appearance ofleaves, results from dissipative instability of the homogeneous in space solution. Westudy nonlinear dynamics and wave propagation of the corresponding reaction-diffusionsystems. Bifurcation of periodic at infinity waves is investigated numerically.

The development of functional-structural plant models has opened interesting perspectivesfor a better understanding of plant growth as well as for potential applications inbreeding or decision aid in farm management. Parameterization of such models is however adifficult issue due to the complexity of the involved biological processes and theinteractions between these processes. The estimation of parameters from experimental databy inverse methods is thus a crucial step. This paper presents some results anddiscussions as first steps towards the construction of a general framework for theparametric estimation of functional-structural plant models. A general family of models ofCarbon allocation formalized as dynamic systems serves as the basis for our study. Anadaptation of the 2-stage Aitken estimator to this family of model is introduced as wellas its numerical implementation, and applied in two different situations: first amorphogenetic model of sugar beet growth with simple plant structure, multi-stage anddetailed observations, and second a tree growth model characterized by sparse observationsand strong interactions between functioning and organogenesis. The proposed estimationmethod appears robust, easy to adapt to a wide variety of models, and generally provides asatisfactory goodness-of-fit. However, it does not allow a proper evaluation of estimationuncertainty. Finally some perspectives opened by the theory of hidden models arediscussed.

In this article a variational reduction method, how to handle the case of heterogenousdomains for the Transport equation, is presented. This method allows to get rid of therestrictions on the size of time steps due to the thin parts of the domain. In the thinpart of the domain, only a differential problem, with respect to the space variable, is tobe approximated numerically. Numerical results are presented with a simple example. Thevariational reduction method can be extended to thin domains multi-branching in 3dimensions, which is a work in progress.

We present a dynamic geometric model of phyllotaxis based on two postulates, primordiaformation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugateare all variations of the same unifying phenomenon and that the difference lies in thechanges in position of initial primordia. We explore the set of all initial positions andcolor-code its points depending on the phyllotactic pattern that arises.