Cell and Developmental Biology, John Innes Centre , Norwich , UK.
Open Biol. 2019 May 31;9(5):190057. doi: 10.1098/rsob.190057. Epub 2019 May 29.
Differential growth is the driver of tissue morphogenesis in plants, and also plays a fundamental role in animal development. Although the contributions of growth to shape change have been captured through modelling tissue sheets or isotropic volumes, a framework for modelling both isotropic and anisotropic volumetric growth in three dimensions over large changes in size and shape has been lacking. Here, we describe an approach based on finite-element modelling of continuous volumetric structures, and apply it to a range of forms and growth patterns, providing mathematical validation for examples that admit analytic solution. We show that a major difference between sheet and bulk tissues is that the growth of bulk tissue is more constrained, reducing the possibility of tissue conflict resolution through deformations such as buckling. Tissue sheets or cylinders may be generated from bulk shapes through anisotropic specified growth, oriented by a polarity field. A second polarity field, orthogonal to the first, allows sheets with varying lengths and widths to be generated, as illustrated by the wide range of leaf shapes observed in nature. The framework we describe thus provides a key tool for developing hypotheses for plant morphogenesis and is also applicable to other tissues that deform through differential growth or contraction.
差异生长是植物组织形态发生的驱动力，在动物发育中也起着基础性作用。尽管通过对组织薄片或各向同性体积的建模已经捕捉到了生长对形状变化的贡献，但在大小和形状的大变化下对各向同性和各向异性体积生长进行建模的框架一直缺失。在这里，我们描述了一种基于连续体积结构的有限元建模的方法，并将其应用于一系列形式和生长模式，为可解析求解的示例提供了数学验证。我们表明，与薄片组织相比，体组织的生长受到更多限制，这减少了通过诸如屈曲等变形来解决组织冲突的可能性。通过各向异性指定生长，可以从体形状生成薄片或圆柱体，其方向由极性场确定。第二个与第一个正交的极性场允许生成具有不同长度和宽度的薄片，这正如自然界中观察到的各种叶片形状所说明的那样。因此，我们描述的框架为植物形态发生的假说发展提供了一个关键工具，也适用于通过差异生长或收缩变形的其他组织。
