Design principles for robust multistability in coupled feedforward-feedback regulatory circuits

Feedforward loops (FFLs) and feedback loops (FBLs) are ubiquitous network motifs that mediate signal filtering, pulse generation, and state switching; yet, how coupling FBLs to FFLs produces robust multistability-a key mechanism for cellular decision-making-remains unclear. Here, we systematically investigate coupled FFL-FBL architectures by focusing on two prevalent FFL types, each with AND or OR logic, yielding four distinct frameworks. For each framework, we enumerate all 3⁶ = 729 possible circuits, corresponding to three possible states (activation, inhibition, or absence) for each of six feedback edges, formulate each circuit as a system of ordinary differential equations, and quantify robustness as the proportion of 100,000 randomly sampled parameter sets exhibiting multistability. Our results reveal two key principles. First, positive self-activation is a primary driver of multistability, but the identity of the critical node(s) depends on the FFL type and logic. Second, coherent FFLs support multistability more readily than incoherent ones, whereas the choice between AND and OR logic has a comparatively weaker effect. Notably, we identify representative high-performing circuits within each framework and find that a small set of circuit designs remain robustly multistable across all four frameworks. These findings advance the theoretical understanding of motif design and provide practical guidelines for engineering synthetic multistable circuits.