Bar-Sinai Yohai, Julien Jean-Daniel, Sharon Eran, Armon Shahaf, Nakayama Naomi, Adda-Bedia Mokhtar, Boudaoud Arezki
Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel.
Racah Institute of Physics, The Hebrew University, Jerusalem, Israel.
PLoS Comput Biol. 2016 Apr 13;12(4):e1004819. doi: 10.1371/journal.pcbi.1004819. eCollection 2016 Apr.
Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.
分化成明确的模式和组织生长被认为是生物体发育的关键过程。然而，尚不清楚这些模式是通过生长被动、均匀地扩张，还是在组织扩张过程中进行重塑。叶片维管网络非常适合研究这个问题，因为叶片近似二维且大小会增长许多倍。在这里，我们通过实验和计算研究叶脉模式如何影响生长。我们首先将生长中的脉管系统建模为一个粘弹性杆网络，并考虑其对外部机械应力的响应。我们使用所谓的纹理张量来量化局部网络几何结构，并揭示生长是不均匀的，类似于复合材料中的非仿射变形。然后，在叶脉分化后，我们对生长中的叶片施加机械力，叶片会通过各向异性生长以及网络在外部应力方向上的重新定向做出响应。与粘弹性杆模型相比，外部机械应力似乎使生长更加均匀。然而，我们通过在杆的厚度中引入随机性以及在杆生长定律中设置一个阈值，使杆具有粘弹塑性，从而使模型与实验数据相协调。总之，我们表明叶脉较高的刚度导致它们沿着外力重新定向，同时生长不均匀性降低。这个过程可能导致叶片对机械应力的强化。更一般地说，我们的工作有助于建立一个框架，通过细胞类型之间机械特性的差异来协调生长和模式。
