The aim of this work is to provide a mathematical model to describe the early stages of the embryonic development of zebrafish posterior lateral line (PLL). In particular, we focus on evolution of PLL proto-organ (said primordium), from its formation to the beginning of the cyclical behavior that amounts in the assembly of immature proto-neuromasts towards its caudal edge accompanied by the deposition of mature proto-neuromasts at its rostral region. Our approach has an hybrid integro-differential nature, since it integrates a microscopic/discrete particle-based description for cell dynamics and a continuous description for the evolution of the spatial distribution of chemical substances (i.e., the stromal-derived factor SDF1a and the fibroblast growth factor FGF10). Boolean variables instead implement the expression of molecular receptors (i.e., Cxcr4/Cxcr7 and fgfr1). Cell phenotypic transitions and proliferation are included as well. The resulting numerical simulations show that the model is able to qualitatively and quantitatively capture the evolution of the wild-type (i.e., normal) embryos as well as the effect of known experimental manipulations. In particular, it is shown that cell proliferation, intercellular adhesion, FGF10-driven dynamics, and a polarized expression of SDF1a receptors are all fundamental for the correct development of the zebrafish posterior lateral line.

这项工作的目的是提供一个数学模型来描述斑马鱼后外侧线（PLL）的胚胎发育的早期阶段。特别是，我们专注于PLL原器官的进化（称原基），从它的形成到开始的周期性行为，直到未成熟的原神经瘤向尾缘的组装，伴随着成熟的原神经瘤在其喙部区域的沉积。我们的方法具有混合积分微分的性质，因为它结合了基于微观/离散粒子的细胞动力学描述和连续化学物质（即基质衍生因子SDF1a和成纤维细胞）空间分布演变的描述生长因子FGF10）。布尔变量改为实现分子受体的表达（即Cxcr4 / Cxcr7和fgfr1）。还包括细胞表型转变和增殖。所得的数值模拟表明，该模型能够定性和定量地捕获野生型（即正常）胚胎的进化以及已知实验操作的效果。特别是，显示出细胞增殖，细胞间粘附，FGF10驱动的动力学以及SDF1a受体的极化表达都是斑马鱼后侧线正确发育的基础。