DataSheet1_Determining the role of advection in patterning by bone morphogenetic proteins through neural network model-based acceleration of a 3D finite element model of the zebrafish embryo.docx

Embryonic development is a complex phenomenon that integrates genetic regulation and biomechanical cellular behaviors. However, the relative influence of these factors on spatiotemporal morphogen distributions is not well understood. Bone Morphogenetic Proteins (BMPs) are the primary morphogens guiding the dorsal-ventral (DV) patterning of the early zebrafish embryo, and BMP signaling is regulated by a network of extracellular and intracellular factors that impact the range and signaling of BMP ligands. Recent advances in understanding the mechanism of pattern formation support a source-sink mechanism, however, it is not clear how the source-sink mechanism shapes the morphogen patterns in three-dimensional (3D) space, nor how sensitive the pattern is to biophysical rates and boundary conditions along both the anteroposterior (AP) and DV axes of the embryo, nor how the patterns are controlled over time. Throughout blastulation and gastrulation, major cell movement, known as epiboly, happens along with the BMP-mediated DV patterning. The layer of epithelial cells begins to thin as they spread toward the vegetal pole of the embryo until it has completely engulfed the yolk cell. This dynamic domain may influence the distributions of BMP network members through advection. We developed a Finite Element Model (FEM) that incorporates all stages of zebrafish embryonic development data and solves the advection-diffusion-reaction Partial Differential Equations (PDE) in a growing domain. We use the model to investigate mechanisms in underlying BMP-driven DV patterning during epiboly. Solving the PDE is computationally expensive for parameter exploration. To overcome this obstacle, we developed a Neural Network (NN) metamodel of the 3D embryo that is accurate and fast and provided a nonlinear map between high-dimensional input and output that replaces the direct numerical simulation of the PDEs. From the modeling and acceleration by the NN metamodels, we identified the impact of advection on patterning and the influence of the dynamic expression level of regulators on the BMP signaling network.

胚胎发育是一种复杂的生理现象，其过程融合了遗传调控与生物力学细胞行为。然而，这些因素对时空形态发生素分布的相对影响尚不明确。骨形态发生蛋白（BMPs）是引导早期斑马鱼胚胎背腹（DV）模式形成的主要形态发生素，而BMP信号通路受到一系列细胞外和细胞内因素的调控，这些因素影响BMP配体的范围和信号。近年来，对模式形成机制的理解取得了进展，支持源汇机制的存在，然而，源汇机制如何塑造三维（3D）空间中的形态发生素模式尚不清楚，形态对胚胎前后（AP）和背腹（DV）轴上的生物物理速率和边界条件的敏感性如何，以及这些模式随时间如何被控制，亦不明确。在整个胚泡化和原肠形成过程中，伴随着BMP介导的DV模式形成，发生了一种被称为上皮外展的主要细胞运动。上皮细胞层在向胚胎的绿色极方向扩散的过程中逐渐变薄，直至完全吞没卵黄细胞。这一动态区域可能通过迁移作用影响BMP网络成员的分布。我们开发了一种有限元模型（FEM），该模型整合了斑马鱼胚胎发育的所有阶段数据，并在生长区域内求解了迁移-扩散-反应偏微分方程（PDE）。我们利用该模型研究上皮外展期间BMP驱动的DV模式形成中的机制。求解PDE在参数探索中计算成本高昂。为了克服这一障碍，我们开发了一种3D胚胎的神经网络（NN）元模型，该模型准确且快速，提供了一种非线性映射，将高维输入与输出联系起来，取代了PDE的直接数值模拟。通过建模和NN元模型的加速，我们确定了迁移对模式形成的影响以及调节因子动态表达水平对BMP信号网络的影响。