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Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

Fig 5

(a) Diffusion permits a bistable gene expression boundary to be accurately placed independently of initial conditions. Left, bifurcation diagram and right, simulation results for Eq (19) with a = 1.7, β = 0.35, D₁ = D₂ = 1, n_a = n_r = 2, K_a = 0.75, K_r = 1, and α₁(x), α₂(x) as shown. Gray shaded region indicates rightward propagation of the front in u₁ (gold) and leftward for the front in u₂ (blue), with lighter shades indicating later time points. Regardless of the initial condition, the boundary propagates toward the point where the toggle switch is “balanced” (the local steady states are such that ). Inverted black triangle indicates the position of a gene expression boundary with the same initial conditions when gene products do not diffuse. For clarity, only u₁ is pictured in the third and fourth plots. (b) Imbalance in gradients or diffusivities may shift the location of the boundary. Right, when D₁ = D₂ (top), the boundary localizes where α₁(x) = α₂(x). Letting D₁ = 2 = 2D₂ (bottom) shifts the localization point in favor of higher u₁. Middle, ODE approximations from the time of boundary formation produce an error less than 0.5% embryo width in predicting boundary location. Simulations are shown for ϵ = 0 (D₁ = D₂) or ϵ = −0.5 (D₁ = 2D₂). Right, changing the diffusion ratio shifts the nullcline for front velocity toward lower concentrations of the faster-diffusing species. Decreasing the diffusivity also decreases the width (increases the steepness) of the respective front (S2 Appendix).

doi: https://doi.org/10.1371/journal.pcbi.1008589.g005