Gene regulatory network (GRN)-based morphogenetic models have recently gained an increasing attention. However, the relationship between microscopic properties of intracellular GRNs and macroscopic properties of morphogenetic systems has not been fully understood yet. Here, we propose a theoretical morphogenetic model representing an aggregation of cells, and reveal the relationship between criticality of GRNs and morphogenetic pattern formation. In our model, the positions of the cells are determined by spring-mass-damper kinetics. Each cell has an identical Kauffman’s $NK$ random Boolean network (RBN) as its GRN. We varied the properties of GRNs from ordered, through critical, to chaotic by adjusting node in-degree $K$ . We randomly assigned four cell fates to the attractors of RBNs for cellular behaviors. By comparing diverse morphologies generated in our morphogenetic systems, we investigated what the role of the criticality of GRNs is in forming morphologies. We found that nontrivial spatial patterns were generated most frequently when GRNs were at criticality. Our finding indicates that the criticality of GRNs facilitates the formation of nontrivial morphologies in GRN-based morphogenetic systems.

中文翻译：

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基因调控网络在形态发生中的重要性

基于基因调控网络 (GRN) 的形态发生模型最近受到越来越多的关注。然而，细胞内 GRN 的微观特性与形态发生系统的宏观特性之间的关系尚未完全了解。在这里，我们提出了一个代表细胞聚集的理论形态发生模型，并揭示了 GRN 的临界性与形态发生模式形成之间的关系。在我们的模型中，单元的位置由弹簧-质量-阻尼器动力学决定。每个单元格都有一个相同的考夫曼 $NK$ 随机布尔网络 (RBN) 作为其 GRN。我们通过调整节点入度将 GRN 的属性从有序、关键到混沌 $K$ . 我们将四个细胞命运随机分配给 RBN 的细胞行为吸引子。通过比较在我们的形态发生系统中生成的不同形态，我们研究了 GRN 的关键性在形成形态中的作用。我们发现，当 GRN 处于临界状态时，最常生成非平凡的空间模式。我们的发现表明，GRN 的重要性有助于在基于 GRN 的形态发生系统中形成重要的形态。