Computer models for understanding plant processes
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The use of computational techniques increasingly permeates developmental biology, from the acquisition, processing and analysis of experimental data to the construction of models of organisms. Specifically, models help to untangle the non‐intuitive relations between local morphogenetic processes and global patterns and forms. We survey the modeling techniques and selected models that are designed to elucidate plant development in mechanistic terms, with an emphasis on: the history of mathematical and computational approaches to developmental plant biology; the key objectives and methodological aspects of model construction; the diverse mathematical and computational methods related to plant modeling; and the essence of two classes of models, which approach plant morphogenesis from the geometric and molecular perspectives. In the geometric domain, we review models of cell division patterns, phyllotaxis, the form and vascular patterns of leaves, and branching patterns. In the molecular‐level domain, we focus on the currently most extensively developed theme: the role of auxin in plant morphogenesis. The review is addressed to both biologists and computational modelers.
I. A brief history of plant models
How far mathematics will suffice to describe, and physics to explain, the fabric of the body, no man can foresee.
D'Arcy Wentworth Thompson (1942, p. 13)
In Book VI of the Enquiry into Plants, Theophrastus (1948; c. 370–285 BCE) wrote: ‘most [roses] have five petals, but some have twelve or twenty, and some a great many more than these’. Although this observation appears to be off by one (Fibonacci numbers of petals, 13 and 21, are more likely to occur than 12 and 20), it represents the longest historical link between observations and a mathematically flavored research problem in developmental plant biology. Numerical canalization, or the surprising tendency of some plant organs to occur preferentially in some specific numbers (Battjes et al., 1993), was described quantitatively in the first half of the 20th century (Hirmer, 1931), analyzed geometrically at the end of that century (Battjes & Prusinkiewicz, 1998), and remains an active area of research. Its genetic underpinnings (e.g. the regulation of ray floret differentiation within a capitulum (Coen et al., 1995; Broholm et al., 2008)) continue to be studied. Less extensive in their historical span, many other links between early observations and current research problems also exist. For instance, several hypotheses attempting to characterize patterns of cell division were formulated in the 19th century (e.g. Errera, 1886) and discussed in the early 20th century (D'Arcy Thompson, 1942, first edition 1917), before becoming the subject of computational studies (Korn & Spalding, 1973), which continue to this day (Nakielski, 2008; Sahlin & Jönsson, 2010; Besson & Dumais, 2011; Robinson et al., 2011).
A broad program of using mathematical reasoning in the study of the development and form of living organisms was initiated almost 100 yr ago by D'Arcy Thompson (1942) in his landmark book On Growth and Form (see Keller, 2002, for a historical analysis). One of his most influential contributions was the ‘theory of transformations’, which showed how forms of different species could be geometrically related to each other. The theory of transformations was extended to relate younger and older forms of a developing organism (Richards & Kavanagh, 1945), but did not incorporate the formation and differentiation of new organs. This limitation was addressed a quarter of a century later by Lindenmayer (1968, 1971), who introduced an original mathematical formalism, subsequently called L‐systems, to describe the development of linear and branching structures at the cellular level. By the mid 1970s, computational models based on L‐systems and other formalisms had been applied to study several aspects of plant development, including the development of leaves and inflorescences, and the formation of phyllotactic patterns (Lindenmayer, 1978). The questions being asked included the impact of distinct modes of information transfer (lineage vs interaction) on plant development, and the relationship between local development and global form. Similar interests underlied the independent pioneering work of Honda and co‐workers on the modeling of trees (Honda, 1971; Borchert & Honda, 1984).