Incoherent modulation of bi-stable dynamics orchestrates the Mushroom and Isola bifurcations

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.