Figure 2.

Tubulation by curved proteins and the bleb tubes fluctuations area exchange. All plots are for $E_{\text{cap}}\gg 1$, as in Eqs. 5 and 6. In each plot, we present ${\varphi }_{\text{t}}^{\text{eq}}$ (black line), ϕ_t in the bleb state (orange line), and ϕ_t + ΔS_b/A (blue line, also proportional to the tension, σ = σ₀ + k_σ(ϕ_t + ΔS_b/A)) as functions of ϕ (see Eq. 5). The vertical gap between the black and orange curves (light orange region) is proportional to the tube-flattened area, and the vertical gap between the blue and black curves (light blue region) is proportional to the stretched area (seized from membrane fluctuations). The dependence on (ε − σ₀s) and on k_σs is indicated by the dashed grid lines (saturation limits) and the ticks that mark the critical density for tubulation, ϕ^∗ (Eq. 6), at equilibrium (black), and in the bleb state (orange). The entropic effects that arise from increasing the temperature (or decreasing all energy parameters proportionally) can be inferred from the different plots. In plots (a1) and (a2), the entropic effects are negligible because $\left(\mathit{\epsilon }-{\sigma }_{0}s\right)\gg 1$ (specifically, (ε − σ₀s) = 30 and k_σs = 50, well within our estimations in Table 1). In plots (b1) and (b2), where (ε − σ₀s) ∼ 1 (specifically, (ε − σ₀s) = 3 and k_σs = 5), one may notice a finite critical density for tubulation, ϕ^∗ > 0, as well as a smoother transition into saturation. The dependence on bleb growth can be inferred from plots (a2) and (b2), where ΔS_b/A is increased to 25% from 5% in plots (a1) and (b1). To see this figure in color, go online.