Movies and VRML files illustrating Figure 6 (b,c). from Volumetric finite-element modelling of biological growth

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（Differential growth）是植物组织形态发生的核心驱动力，同时在动物发育过程中也发挥着基础性作用。尽管已有研究通过组织片层（tissue sheets）或各向同性（isotropic）体积建模，刻画了生长对形态改变的贡献，但目前仍缺乏能够在三维空间中，针对尺寸与形态的大幅变化，同时对各向同性与各向异性（anisotropic）体积生长进行建模的框架。本文提出了一种基于连续体积结构有限元建模（finite-element modelling）的方法，并将其应用于多种形态与生长模式，同时针对存在解析解（analytic solution）的示例提供了数学验证。本研究表明，片层组织与实体组织（bulk tissues）间的核心差异在于：实体组织的生长受到更强约束，从而降低了通过屈曲（buckling）等变形实现组织冲突消解的可能性。通过由极性场（polarity field）定向的各向异性指定生长，可从实体形态生成组织片层或圆柱体结构。增设一个与首个极性场正交的第二极性场，即可生成具有不同长宽的片层结构，这一点可通过自然界中广泛存在的多样叶片形态得到例证。综上，本文所提出的框架为植物形态发生的假说构建提供了关键工具，同时也适用于其他通过差异生长或收缩（contraction）发生形变的组织。