Fig. 4.

Dynamical behavior of lumen growth. Dynamical behavior of the normalized lumen height $h\left(t\right)/L$, junctional extension $r_l\left(t\right)/L$, and angle $\theta \left(t\right)/\pi$ are shown as a function of normalized time $t/\tau$ (lower abscissa) and time in hours (upper abscissa), where $\tau =2\times 10^{-8}\mathit{\text{n.u.}}$ is the cortex time (assumed 1,000 s). Changing pump efficiency $\delta {\rho }_i$ shows three different characteristic behaviors: (A) Overdamped evolution toward steady state at $\delta {\rho }_i=1.0\times 10^8\mathit{\text{n.u.}}$, (B) underdamped evolution toward steady state at $\delta {\rho }_i=1.35\times 10^8\mathit{\text{n.u.}}$, and (C) sustained oscillations $\delta {\rho }_i=1.38\times 10^8\hspace{0.17em}\mathit{\text{n.u.}}$ The numerics have been obtained for values of ${\xi }_V=0.49$, ${\xi }_i=0.50$, ${\mathrm{\Lambda }}_v=1\mathit{\text{n.u.}}$, $\mathrm{\Lambda }={\mathrm{\Lambda }}_i{k}_{B}T\tau /{\rho }_{cell}L=1.1\times 10^8\mathit{\text{n.u.}}$, ${\sigma }_0=10^7\mathit{\text{n.u.}}$, $\delta {\rho }_{ext}=-2\times 10^6\mathit{\text{n.u.}}$, and ${\rho }_{cell}=10^9\mathit{\text{n.u.}}$