Dynamics and steady states of the MHB model. (A) Top view ofsteady state expression pattern of the six IsO genes Otx2, Gbx2, Wnt1,Fgf8, En (subsuming En1 and En2) and Pax(subsuming Pax2 and Pax5) in the MHR around the MHB (blackarrow head) of the mouse embryo at E10.5. The midline represents theanterior-posterior axis of the embryo. (B) Dynamics of the approximatedsteady states for the parametersα
= 1 andβ
= 1 fori = 1,…,8, all Hill interaction parameters areidentical with n = 2 and k = 0.1.Here the sum of gene and morphogen, i.e. Fgf8 plus Fgf8 andWnt1 plus Wnt1, is plotted for better visibility. Both steadystates show a sharpening of the En and Pax domain marked bythe dashed circles. The second steady state shows a sharp Fgf8boundary indicated by the dashed circle. Both simulations were run forT = 0 to T = 50, shown here areT = 0 (first panel), T = 1 (secondpanel) and T = 50 (third panel) for a grid of sizeN = 1,000. At T = 50 we found thatthe system has reached a steady state. (C) Enlarged view of thesharpening, i.e. discontinuities, in the En expression level arisingin both steady state. (D) Enlarged view of the sharpening, i.e.discontinuities, in the Fgf8 expression level arising only in thesecond steady state. Abbreviations: MHB: mid-hindbrain boundary, Mb: midbrain,Hb: hindbrain.