Near-critical gene expression in embryonic boundary precision

In the morphogenesis of the fruit fly embryo, a key process is the longitudinal expression of the hunchback (hb) gene in response to the bicoid (bcd) morphogen gradient. The spatial profile of hb expression is characteristically a steep step, with the boundary being extraordinarily precise, showing little variance across samples. While the formation of this boundary has been proposed to occur at a dynamical critical point, evidence suggests that it is more likely to occur in a bistable regime. To address this gap in our understanding, we develop a minimal model for the expression of hb that accounts for its cooperative activation by bcd and its own positive autoregulation. We identify a single parameter that tunes the system from its monostable regime to its bistable regime, crossing the critical point in between. We find that boundary precision is maximized when the system is weakly bistable—near, but not at, the critical point—optimally negotiating the trade-off between two key effects of bistability: sharpening the boundary and amplifying its noise. Further, incorporating the diffusion of hb proteins into our model, we show that boundary precision is maximized simultaneously at an optimal degree of bistability and an optimal diffusion strength. Our work elucidates design principles for precise boundary formation and has general implications for pattern formation in multicellular systems.