Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries
Fig 4
(a) In the posterior of the embryo, the Hb boundary behaves as a bistable propagating wavefront. Simulations (left, middle) and bifurcation diagrams (right) of the Hb patterning system with a static Bcd gradient show that the Hb gene expression boundary becomes a propagating bistable wavefront toward the posterior of the embryo (gray shaded region) and would continue to propagate slowly thereafter (thin lines) if not for external influences such as downstream genetic networks, cellularization, or gastrulation. Blue indicates wild-type (WT) and gold, stau- mutant. Right, the Bcd gradient for stau- (gold dotted line) has lower amplitude than that of the WT Bcd gradient (blue dotted line), which shifts the bifurcation (critical) point anteriorly. The local front velocity calculated from Eq (13) rapidly decreases posterior to the critical point but remains strictly positive, indicating that a bistable gene expression boundary will propagate posteriorly. The Bcd1.0 bifurcation diagram (see (b)) is almost identical to that of stau-. Top insets compare boundary location from PDE simulations (solid black) to the location from an ODE approximation (red dashed) derived from the local front velocities, with ϵ equal to a quarter of the boundary width. Simulation start t = 0 is 10 min before the beginning of nc14. (b) Decay in the Bcd gradient shifts the bistable region anteriorly over time, causing boundaries to stabilize at slightly anterior positions. Bottom, there is a greater discrepancy between boundaries forming over dynamic (solid lines) vs. static (dash-dot line) in the WT (blue) and Bcd1.0 (red) mutant than the stau- (gold) mutant due to differences in front emergence. Specifically, homogeneous initial conditions below the maximum value of the unstable steady state cause the boundary in the stau- mutant to appear within the bistable region, such that boundary propagation continues at roughly the same speed regardless of gradient dynamics. In contrast, in WT and Bcd1.0 the boundary is preinduced anterior to the critical point. A decrease in morphogen concentration in time causes the boundary to reach the critical point sooner, but also reduces the total time spent by the boundary in the region of rapid propagation, resulting in more anterior placement relative to the static case. All parameters are as given in [40].