A "morphogenetic action" principle for 3D shape formation by the growth of thin sheets

How does growth encode form in developing organisms? Many
different spatiotemporal growth profiles may sculpt tissues into the same
target 3D shapes, but only specific growth patterns are observed in animal
and plant development. In particular, growth profiles may differ
in their degree of spatial variation and growth anisotropy, however, the
criteria that distinguish observed patterns of growth from other possible
alternatives are not understood. Here we exploit the mathematical
formalism of quasiconformal transformations to formulate the problem of
``growth pattern selection'' quantitatively in the context of 3D
shape formation by growing 2D epithelial sheets. We propose that
nature settles on growth patterns that are the `simplest' in a
certain way. Specifically, we demonstrate that growth pattern
selection can be formulated as an optimization problem and solved for the
trajectories that minimize spatiotemporal variation in areal growth rates
and deformation anisotropy.  The result is a complete prediction
for the growth of the surface, including not only a set of intermediate
shapes, but also a prediction for cell displacement along those surfaces
in the process of growth. Optimization of growth trajectories for
both idealized surfaces and those observed in nature show that relative
growth rates can be uniformized at the cost of introducing
anisotropy. Minimizing the variation of programmed growth rates
can therefore be viewed as a generic mechanism for growth pattern
selection and may help to understand the prevalence of anisotropy in
developmental programs.

发育中的生物体如何通过生长编码形态？诸多不同的时空生长模式可将组织塑造为相同的目标三维形态，但在动植物发育过程中仅会观测到特定的生长模式。具体而言，不同生长模式的空间变异程度与生长各向异性存在差异，但目前仍未明确区分观测到的生长模式与其他潜在可选模式的评判标准。本研究借助拟共形变换（quasiconformal transformations）的数学形式框架，针对二维上皮细胞层（2D epithelial sheets）生长形成三维形态的场景，定量构建了“生长模式选择”问题。我们提出，自然界所优选的生长模式在特定意义上具备“最简性”。具体而言，我们证明生长模式选择可被表述为一个优化问题，并可求解出同时最小化面积生长速率与变形各向异性的时空变异的生长轨迹。该结果可完整预测该表面的生长过程，不仅能给出一系列中间形态，还可预测生长过程中细胞沿该表面的位移情况。对理想化表面与自然界中观测到的表面的生长轨迹进行优化后均表明，相对生长速率可通过引入各向异性来实现均匀化。因此，最小化程序化生长速率的变异可被视为一种通用的生长模式选择机制，或有助于理解发育程序中各向异性的普遍存在。