In biological networks, steady state dynamics of cell-fate regulatory genes often exhibit Mushroom and Isola kind of bifurcations. How these complex bifurcations emerge for these complex networks, and what are the minimal network structures that can generate these bifurcations, remain elusive. Herein, by employing Waddington’s landscape theory and bifurcation analysis, we demonstrate that Mushroom and Isola bifurcations can be realized with four minimal network motifs that are constituted by combining a positive feedback motif with various incoherent feed-forward loops. Our study reveals that the intrinsic bi-stable dynamics originating from the positive feedback motif can be fine-tuned by altering the extent of the incoherence of these minimal networks to produce these complex bifurcations. These modeling insights will be useful in identifying the possible network motifs that may give rise to either Mushroom or Isola bifurcation in other biological systems.

在生物网络中，细胞命运调节基因的稳态动力学通常表现出蘑菇和 Isola 类型的分叉。这些复杂网络如何出现这些复杂的分叉，以及可以生成这些分叉的最小网络结构是什么，仍然难以捉摸。在此，通过采用 Waddington 的景观理论和分岔分析，我们证明了 Mushroom 和 Isola 分岔可以通过四个最小网络模体来实现，这些网络模体是通过将正反馈模体与各种不连贯的前馈循环相结合而构成的。我们的研究表明，可以通过改变这些最小网络的不连贯程度来微调源自正反馈基序的内在双稳态动力学，以产生这些复杂的分叉。