Self-assembly of polyhedral bilayer vesicles from Piezo ion channels

Piezo ion channels underlie many forms of mechanosensation in vertebrates and have been found to bend the membrane into strongly curved dome shapes. We develop a methodology describing the self-assembly of lipids and Piezo proteins into polyhedral bilayer vesicles. We validate this methodology for bilayer vesicles formed from bacterial mechanosensitive channels of small conductance, for which experiments found a polyhedral arrangement of proteins with snub cube symmetry and a well-defined characteristic vesicle size. On this basis, we calculate the self-assembly diagram for polyhedral bilayer vesicles formed from Piezo proteins. We find that the radius of curvature of the Piezo dome provides a critical control parameter for the self-assembly of Piezo vesicles, with high abundances of Piezo vesicles with octahedral, icosahedral, and snub cube symmetry with increasing Piezo dome radius of curvature.

Figure 1

Schematic of MPPNs formed from Piezo ion channels. The thick gray curve shows the Piezo dome with radius of curvature $R_D$ and cap angle $\alpha$. The blue curve shows the membrane footprint of the Piezo dome with arclengths $s=0$ and $s=s_b$ at the inner and outer membrane footprint boundaries, respectively. At $s=s_b$, the Piezo membrane footprint is assumed to connect smoothly to the membrane footprints associated with the neighboring Piezo proteins on the MPPN surface, with contact angle $\beta$. We denote the inner and outer membrane footprint boundaries along the $r$-axis by $r_0$ and $r_b$, respectively. The MPPN radius is given by $R=r_b/sin\beta$. For each $s_b$, the Piezo membrane footprint is completely determined by a given set of values of $\alpha , R_D, r_0=R_{D}sin\alpha$, and $\beta$, which are indicated in gray. For simplicity, we only show here one of the $n$ proteins on the MPPN surface.

Figure 2

MPPN self-assembly diagrams for MPPNs formed from MscS obtained using (a) the arclength parametrization of Eq. (7) and (b) the Monge parametrization of Eq. (7). The horizontal axes show the bilayer-protein contact angle, $\alpha$, and the vertical axes show the protein number fraction in solution, $c$. The dominant $n$-states of MPPNs are indicated by integers. The white dashed curves show transitions in the dominant MPPN $n$-states, with the colors indicating the maximum $\varphi \left(n\right)$ among all MPPN $n$-states considered. We use the same color bar in panels (a) and (b). The dashed horizontal lines indicate the parameter values $c\approx 7.8\times {10}^{-8}$ and $\alpha \approx 0.46$–0.54 rad corresponding to experiments on MPPNs formed from MscS [16, 17], in which the snub cube with $n=24$ MscS proteins was found to provide the dominant MPPN symmetry (models of the snub cube below $n=24$).

Figure 3

Fractional abundance of MPPN $n$-states, $\varphi \left(n\right)$, versus bilayer-protein contact angle, $\alpha$, for selected (dominant) MPPN $n$-states in Fig. 2 (indicated by integers) obtained with the arclength parametrization of Eq. (7) and the protein number fraction $c\approx 7.8\times {10}^{-8}$ used for the MPPN self-assembly experiments in Refs. [16, 17]. As in Fig. 2, we calculated all curves by interpolation with respect to $\alpha$ of numerical results at a resolution $\mathrm{\Delta }\alpha =0.01$ rad [dots in panel (a)], using third-order splines. Panel (a) compares these interpolations with the corresponding results obtained at a finer resolution $\mathrm{\Delta }\alpha =0.002$ rad (crosses). Panel (b) compares the curves in panel (a) obtained using $A=A_D$ in Eq. (27) (solid curves) with the corresponding results obtained using $A=A_S$ (dashed curves).

Figure 4

MPPN self-assembly diagram for MPPNs formed from Piezo ion channels, obtained from the arclength parametrization of Eq. (7), as a function of the radius of curvature of the Piezo dome, $R_D$, and the protein number fraction in solution, $c$. Colors indicate the maximum $\varphi \left(n\right)$ among all MPPN $n$-states considered. The color bar employed here is identical to the color bar employed in Fig. 2. As in Fig. 2, the dominant $n$-states of MPPNs are indicated by integers, and the white dashed curves show transitions in the dominant MPPN $n$-states. The horizontal dashed line indicates the protein number fraction $c\approx 7.8\times {10}^{-8}$ used in experiments on MPPNs formed from MscS [16, 17], while the vertical dashed line shows the Piezo dome radius of curvature $R_D\approx 10.2$ nm observed for a closed state of Piezo ion channels [10]. The polyhedra models illustrate dominant MPPN symmetries predicted by the MPPN self-assembly diagram with, from left to right, $n=6$ (octahedron), $n=12$ (icosahedron), and $n=24$ (snub cube). Gray shading indicates the region of the MPPN self-assembly diagram dominated by MPPN $n$-states with $n=80$, which may be a spurious result of our constraint $3\le n\le 80$.

Figure 5

Fractional abundance of MPPN $n$-states obtained from the arclength parametrization of Eq. (7), $\varphi \left(n\right)$, versus radius of curvature of the Piezo dome, $R_D$, for selected (dominant) MPPN $n$-states in Fig. 4 at the protein number fraction $c\approx 7.8\times {10}^{-8}$ used in experiments on MPPNs formed from MscS [16, 17]. As in Fig. 4, the vertical dashed line shows the Piezo dome radius of curvature $R_D\approx 10.2$ nm observed for a closed state of Piezo ion channels [10]. All curves were obtained through interpolation of $\varphi \left(n\right)$ with respect to $R_D$ for numerical results at a resolution $\mathrm{\Delta }{R}_D=0.2$ nm (dots), using third-order splines.