Assessing the parameter optimisations.

(A) A plot of the fitness improvements (decrease in shape difference on the y-axis) against the successive iterations of the optimisation process (x-axis). At each iteration the shape difference decreases until it converges to a stable minimum. For comparison, the green line shows the difference between S_t1 and pS_t1 when using the BrdU/IddU data (i.e. not optimised). The red line shows the result of the optimisation with only positive proliferation (Figure 5B–E), which achieves a lower shape difference. The blue curves are the results of optimisations which allowed negative s values (tissue shrinkage). The multiple blue lines represent optimisations starting with different initial proliferation patterns, and all converge to similar results which are ∼70% lower (better) than the BrdU/IddU data (green line). Although the optimal shape difference was the minimal difference, we have illustrated the global optimum of the landscape as a high peak, following the convention of the hill-climbing analogy. Starting from different points of the fitness landscape, the parameter optimisation converges to the same “mountain peak” with the same basic pattern of growth values. In (B) and (C) the x-axis is the same as (A), but the y-axis plots the changing s values for different positions within the limb bud. Neighbouring pairs of vertices are tracked for each of four regions (distal, central, dorsal, and ventral) and illustrate that although the two cases start with different distributions of initial s values, these parameters converge to the same general layout, with high growth in distal regions, low in central regions, and tissue shrinkage in dorsal and ventral regions. (D–G) shows the results in more detail for one of the optimisation runs. The final growth pattern (D) clearly shows a discrete region of very high proliferation at the distal tip (red/yellow) and shrinking areas dorsal and ventrally (blue). The resulting tissue displacements (E) generate a new shape pS_t1 (green surface in F,G) which is much flatter than before and shows a close correspondence to the real shape S_t1 (blue in G). Limb orientations are as shown in Figure 2F.