Growth-Algorithm Model of Leaf Shape

David A. Young Lawrence Livermore National Laboratory Mail Stop L-45 7000 East Avenue Livermore, California 94550 email address: young5@llnl.gov April 2010

ABSTRACT The innumerable shapes of plant leaves present a challenge to the explanatory power of biophysical theory. A model is needed that can produce these shapes with a small set of parameters. This paper presents a simple model of leaf shape based on a growth algorithm, which governs the growth rate of leaf tissue in two dimensions and hence the outline of the leaf. The growth of leaf lobes is governed by the position of leaf veins. This model gives an approximation to a wide variety of higher plant leaf shapes. The variation of leaf shapes found in closely related plants is discussed in terms of variability in the growth algorithms. The model can be extended to more complex leaf types.

2 I. INTRODUCTION

Modern physics has recently incorporated the problem of complex natural phenomena as a strong research focus. This has brought problems of biological process and structure into the domain of theoretical physics. Biological pattern formation or morphogenesis is an important aspect of complexity theory. One example of pattern formation is the origin of leaf shapes. Human admiration of the marvelous variety of leaf shapes is ancient, and has stimulated modern scientific analysis of how leaves attain their shapes. Experimental investigations on the causation of leaf shapes began more than a century ago, and continue today [1]. Genetic components of leaf growth have been discovered and are beginning to provide a picture of how gene products influence leaf shape [2].
However, it is a legitimate question how far molecular-genetic investigations can go in analyzing morphogenetic processes, which inevitably involve multi- cellular organization. Like other morphogenetic processes, leaf growth and shape are poorly understood. Theoretical studies are valuable in providing models of pattern formation that stimulate new research and that can be tested against experimental data [3,4]. Theory offers the promise of bridging the very large gap between the expression of genes and the final shape of an organ. There are a number of divergent hypotheses explaining leaf form. One of the earliest is the work of Thompson in his famous book, On Growth and Form. In a brief section on leaf morphogenesis, Thompson suggests that a polar coordinate function fit to a leaf outline indicates a vector diagram of the growth process [5]. He gives an example of a function that resembles the horse chestnut (genus Aesculus) leaf with its many lobes. Other models include the Lindenmayer L-system [6], fractal analysis [7], a Turing reaction-diffusion process [8], an iterative space-filling branching process [9], and a linear force-relaxation model [10]. The formation of the vein network of leaves, which is closely related to leaf growth, has also been studied in recent theoretical work [11,12]. Much more experimental and theoretical work is needed to develop a convincing theory of leaf shape. In this paper I introduce a very simple model of leaf growth that produces a spectrum of shapes approximating those observed in nature. My objective is to explain not only the range of leaf shapes found in the higher plants, but also the large variations in shape seen in leaves on closely related plants. The results of the model simulations naturally lead to speculations about the combined influence of genes and environment on leaf morphogenesis.

3 II. LEAF SHAPES IN NATURE

Leaves have evolved over millions of years to optimize light collection, transport of nutrients to and from the plant body, and mechanical stability against natural stresses. Even under the optimizing force of natural selection, however, leaves are found in innumerable forms, showing that leaf shape is a response to multiple competing influences. Variability of leaf shape within a single genus or even on a single plant is especially interesting, because it indicates that the controlling shape-generating “algorithm” has multiple components that can be independently varied. The study of complex physical structures such as diffusion-limited aggregates, dendritic crystals, etc., has led to the conclusion that they are the result of endlessly iterated simple processes [13].
It is plausible that biological structures can be explained similarly, and that only a few model parameters are needed to generate the variety of structures being considered. The task of the theorist is to decipher the structure-generating algorithm from the various shapes that appear in nature. Botanists have developed numerous terms for the shapes of leaves as an aid to identifying plant species [14], but these names are not based on a biophysical understanding of leaf growth. It is perhaps now time for

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