Fig. 6.

Gradients produced by a gating-electrodiffusion mechanism are stable, essentially independent of size scale, and reach similar steady-state values which are ideal for gene regulation. Emergent $V_{mem}$ gradients produced on three different model sizes (A) spontaneously arise for a regulatory system involving a gap junction permeable anion, which inhibits a K⁺ leak channel to generate a self-reinforcing spatial feedback (B). The electrodiffusing anion also inhibited canonical Wnt signaling (e.g. via ATP⁴⁻ inhibition of Wnt signaling through CHD8 (Thompson et al., 2008; Nishiyama et al., 2012)). Canonical Wnt/$\beta$-Cat signaling then inhibited ERK signaling to define both Wnt and ERK signaling gradients (C) consistent with head and tail anatomical domains in both whole worms of diverse size and multiple cut fragments.