Theoretical models for branch formation in plants

Abstract

Various branch architectures are observed in living organisms including plants. Branch formation has traditionally been an area of interest in the field of developmental biology, and theoretical approaches are now commonly used to understand the complex mechanisms involved. In this review article, we provide an overview of theoretical approaches including mathematical models and computer simulations for studying plant branch formation. These approaches cover a wide range of topics. In particular, we focus on the importance of positional information in branch formation, which has been especially revealed by theoretical research in plants including computations of developmental processes.

Keywords: Branch; Divarication; Mathematical model; Plant morphogenesis; Theoretical approach.

Fig. 1

Diversity of branches. a–d Development of diversity in divarication: from left to right, disk-like architectures grown with equally spaced periodic patterns, as described by Harrison and Kolář (1988), Holloway and Harrison (1999), and Nakamasu et al. (2014). Branches gradually develop during growth processes. a No side branches, b, c bifurcation, and d monopodial branching. As branch development proceeds, the branches tend to overlap. c, d Scale-downs of iteratively added units to avoid collision is included. e–j Representations of three-dimensional branching with particular branching rules were generated based on Honda’s I-model (Borchert and Honda 1984); e, h bifurcation; divergent angle is 90° and branch angle of two daughter branches is 45°; f, i alternate phyllotaxis; divergent angle is 137.5° and branch angle is 45°; and g, j opposite phyllotaxis; branch angles of two lateral branches are 45° and divergent angle is 90°. e–g Branch lengths are the same for the whole tree. h–j Branch lengths decrease dependent on the branch hierarchies with ratio 0.8

Fig. 2

Interactions with the external environment in branch formation. The formation of branch architecture in plants is considered to be produced by changes in growth intensity, which seem to include activation (a, c) and inhibition (b, d) of growth. Each case includes open boundaries (a, b) and closed boundaries (c, d). With open boundaries, branches can interact via the external environment (a, b). By contrast, with closed boundaries, the generated branches need to avoid overlaps (c, d). In some cases, even branches with closed boundaries can be modified by the local environment. e A crown shyness-like phenomenon observed in camphor trees, which is considered to be an example of the local interaction

Fig. 3

Deformations of circumference in three-dimensional spaces. a A leaf primordium in a shoot apical meristem of Eschscholzia californica Cham.; scale bar 100 µm. b Divarication generated by deformation of a ring on a two-dimensional plane. The ring was grown based on an equally spaced periodic pattern (based on the model in Nakamasu et al. 2014). c Deformations of divarication on a two-dimensional plane in a three-dimensional space. a The boundary between the adaxial and abaxial sides of the primordium is outlined with a white dashed line. Continuous deformation of the boundary is considered to correspond to the deformation of a ring on a two-dimensional plane

Fig. 4

Longitudinal sections of virtual broccoli inflorescences. Each row shows a virtual broccoli inflorescences and its section generated by the same branch rules shown in Fig. 1h–j. a–c Bifurcation, d–f alternate phyllotaxis and g–i opposite phyllotaxis with lateral branching based on Honda’s I-model (Borchert and Honda 1984). d–f The middle row shows a similar model to actual broccoli (the divergence angle is 137.5°). b, e, h Each broccoli inflorescences is dissected through the gray plane. c, f, i The expected cross-sections of each broccoli inflorescences

Fig. 5

Different modes of branching. Contrasting modes of divarication observed in leaf shapes in closely related ferns; a bifurcated Microsorum pteropus Copel. var. windelov and d laterally branched Microsorum sp. ‘Fork leaf’, based on Miyoshi et al. (2019). Scale bar 2 cm. b, c, e, f Three-dimensional diversity in each mode. b, c Bifurcation and e, f lateral branching. Each branch was sectioned horizontally in the gray plane. The sections of each layer (L1–L3) is shown beneath each branch