Roger V. Jean

Symmetry in Plants

Foreword - the aims and scope of the book a glance at each chapter, D. Barabe and R.V. Jean prologues - by a botanist, R.O. Erickson, by a mathematician, I. Adler, by a crystallographer, A. Mackay, by a molecular geneticist, A. Lima-de-Faria part 1 - basic information gathering in phyllotaxis - data-experimentation part 2 - pattern recognition in phylotaxis - description part 3 pattern generation in phyllotaxis - modelling part 4 origins of phyllotaxis - homology-comparative morphology-exotic phyllotaxis, literature, appendices.

A systemic model of growth in botanometry

Abstract This paper deals with the problem of phyllotaxis. It is well known that each type of phyllotactic arrangement of the primordia of plants leads to a recurring series of positive integers 〈 H ( k )〉, where H ( k ) = H ( k −1)+ H ( k −2), closely related to a precise divergence angle between the primordia. We already introduced, in former articles, a mechanism using a concept of information, a bi-dimensional architectural concept of hierarchy, a principle of optimal design and specific languages to cope with the problem, and we drew preliminary conclusions. This setting is enhanced here as it is shown to give the different types of phyllotaxis, to explain their relative occurrences in Nature and to suggest verifiable predictions.

Introductory review: Mathematical modeling in phyllotaxis: the state of the art

Abstract Phyllotaxis is one of the most striking and puzzling characteristics of apical activity. In the last decade many mathematical models have been put forward of the theories dealing with the problem of the origin of these patterns displayed by the primordia of plants. This paper brings these models together in order to underline their respective successes, to compare them, and to suggest directions for research towards a more complete solution of the problem, that is, towards a better integration of its many facets.

Growth and entropy: Phylogenism in phyllotaxis

Abstract Despite many very recent advances concerning the fundamental problem of phyllotaxis, to explain why the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, …) arises in the secondary spirals on plants, the domain of phyllotaxis is still seeking a rational explanation. All the known theories (Veen, Adler, Thornley, Richards, …: inhibitor, space-filling, contact pressure, mechanistic, …) try to predict the place of each primordium in the apex, on the basis of chemical or mechanical forces or fields; they all have their obvious exceptions. To take into account all the forces and gradients involved in the distal zone of the apex, from where the primordia are emerging, we must consider the problem as phylogenetic (evolution of species). We thus introduce a concept of entropy (from the Greek, evolution) in the domain of growth, by means of the new concept of relational tree, each type of phyllotaxis being represented by a tree. The latter concept is born from theoretical considerations but it is supported by botanical works (Bolle, Berdyshev, Woodger, …) and the former concept has been inspired by the works of Collot in bio-thermodynamics. In the conceptual framework thus obtained, we have proved, as an application of the Principle of Optimal Design, that the cost (the entropy) of normal asymmetric phyllotaxis (characterized by the Fibonacci sequence) is minimal among all other types of phyllotaxis. We have presented this development using the four kinds of axioms found in axiomatic mathematical physics.

A basic theorem on and a fundamental approach to pattern formation on plants

Abstract This article deals with a theorem at the basis of descriptive phyllotaxis, and with an explanatory approach in causal phyllotaxis. The theorem establishes the relation between the divergence angle and the visible opposed spiral pairs observed in structures such as proteins, meristematic apices, and the surfaces of plants. The lack of such a theorem has been an important source of errors, confusions, and obstructions in the mathematical approaches as well as in the biological interpretation of phyllotaxis. An elementary though not simple proof of it is given here together with an intuitive, less technical presentation. Many applications are mentioned. It is shown that the theorem gives a new meaning to Bolles theory of bifurcated induction lines, which in turns throws new light on what is called the hierarchical representation of phyllotaxis. An explenatory model of phyllotaxis, in which the theorem plays an important role, is constructed in this representation. The idea of entropy, on which the model is based, is seen to constitute a most fundamental approach to phyllotactic systems.

Phyllotaxis — The Way Ahead, A View on Open Questions and Directions of Research

Suggestions are made for future research in the multidisciplinary subject of phyllotaxis, and various promising viewpoints for the study of the phenomenon are pointed out.

An L-system approach to nonnegative matrices for the spectral analysis of discrete growth functions of populations

Abstract The author shows that Perron-Frobenius theorem is a valuable tool for the investigation of L-systems, as he deepens the analysis of growth functions of cell populations with lineage control, by means of suitable results proposed hereinafter. Among these results are conditions for ƒ(t),t=1,2,3,… , to be asymptotically equal to brt, where b is a constant and ƒ(t) is the growth function of the system generated by the nonnegative irreducible square matrix C with spectral radius r. The values of lim C t r r , lim ƒ(t+1) ƒ(t) , and lim ƒ(t) r t are given in terms of the coefficients of the eigenvalues of C in the analytical expression for ƒ(t). The technique of growth functions is shown to be an appropriate tool for the analysis of the above type of matrices, which are relevant in many fields of application.

A rigorous treatment of Richards's mathematical theory of phyllotaxis

Abstract This paper is devoted to clarifying definitively the mathematical relations between the parameters involved in Richardss centric representation of the spirally arranged primordia of plants. The parameters are the divergence angle, plastochrone ratio, and phyllotaxis index, together with the necessary adjustments on the approximately conical surface of the apex; the spirals (the so-called parastichies) are logarithmic. We give examples and discuss, in the light of recent advances, some trials dealing with Richardss descriptive theory.

On the allometric growth of tissues in fruits

This paper deals with the relative growth of three different fruit tissues. Their morphogenetic periods and the mathematical constraints involved are described, and more precisely, the paper shows an allometric relationship (Y=nX m ) between the widths (X, Y) of the main tissues in stone fruits such as cherries, peaches and prunes. The mathematical relationships between the growth of the mesocarp and of the endocarp of somePrunus fruits are described, and it is proved that before the formation of the embryo, growth is allometric, in agreement with conclusions drawn from some experimental data. However, according to another study, the growth of the mesocarp and of the endocarp are ruled by autocatalytic and monomolecular functions, before as well as after the formation of the embryo. In this case, it is proved that if allometry exits in stone fruits, it can only be anantiometry (m=−1). To solve the dilemma, two main alternatives are proposed and discussed. We conclude that, while allometry is established on reasonable grounds before the formation of the embryo, after the formation of the embryo the mesocarp and endocarp evolve independently since a center for the coordination of growth no longer exists, and each tissue can grow according to its own independent rules.

An interpretation of Fujita's frequency diagrams in phyllotaxis

Fujitas diagrams in phyllotaxis, showing the frequencies of divergence angles as a function of these angles for low phyllotactic patterns such as (2, 1) and (3, 2), which are approximately normal curves centered at the limitdivergence angle of 137.51°, are shown to be puzzling when compared to results and observations in the field. An analysis of these diagrams is proposed, in the context of Fujitas methodology, of data from other sources, of a mathematical theorem on lattices, and of the contact pressure theory of phyllotaxis.