Symmetry and Size of Membrane Protein Polyhedral Nanoparticles

In recent experiments [T. Basta et al., Proc. Natl. Acad. Sci. U.S.A. 111, 670 (2014)] lipids and membrane proteins were observed to self-assemble into membrane protein polyhedral nanoparticles (MPPNs) with a well-defined polyhedral protein arrangement and characteristic size. We develop a model of MPPN self-assembly in which the preferred symmetry and size of MPPNs emerge from the interplay of protein-induced lipid bilayer deformations, topological defects in protein packing, and thermal effects. With all model parameters determined directly from experiments, our model correctly predicts the observed symmetry and size of MPPNs. Our model suggests how key lipid and protein properties can be modified to produce a range of MPPN symmetries and sizes in experiments.

Figure 1

Schematic of MPPNs [4]. (a) MscS are embedded in a lipid bilayer with the MscS cytoplasmic region outside MPPNs [4]. (b) Lipid bilayer bending deformations (blue curves) induced by the observed MscS structure [3, 5, 6] with protein radius ${\rho }_i$ and bilayer-protein contact angle $\alpha$. The membrane patch radius ${\rho }_o$ and boundary angle $\beta$ are determined by the MPPN size and the number of proteins per MPPN. (MscS Protein Data Bank ID 2OAU with [8] different colors indicating different MscS subunits).

Figure 2

Minimized total MPPN energy $E_{\min }$ for MPPNs formed from MscS [4, 7] with the contributions $E_b$ and $E_d$ due to bending deformations and packing defects versus $n$ at $\alpha =0.5\mathrm{rad}$. The inset shows the optimal sphere coverage $p(n)$ for $n$ identical nonoverlapping circles [43] with particularly favorable packings at $n=12$ (icosahedron), $n=24$ (snub cube), and $n=48$. (Inset after Ref. [43]).

Figure 3

Front and side views of the minimum-energy MPPN configuration obtained from our minimal molecular model of MPPN organization. The larger and smaller disks represent membrane proteins and the lipid bilayer, with disk sizes corresponding to [16] MscS [6] and diC14:0 lipids [40], respectively. The green and blue lines are obtained by connecting the centers of nearby MscS in the simulated MPPN configuration and by fitting the simulated MPPN configuration to a snub cube (dextro) using least-square minimization.

Figure 4

MPPN self-assembly phase diagram obtained from Eq. (4) as a function of protein number fraction $c$ and bilayer-protein contact angle $\alpha$. The color map in the upper panel shows the maximum values of $\varphi (n)$ associated with the dominant $n$-states of MPPNs. The dominant $n$ are indicated in each portion of the phase diagram, together with the associated MPPN symmetry [43]. Black dashed curves delineate phase boundaries. The white dashed line indicates the value $c\approx 7.8\times {10}^{-8}$ corresponding to experiments on MPPNs [4] and the $\alpha$-range associated with MscS [3, 6, 16]. The lower panel shows $\varphi (n)$ for $n=20$, 22, 24, 27, and 30 as a function of $\alpha$ along the white dashed line in the upper panel. We use the model parameter values ${\rho }_i\approx 3.2\mathrm{nm}$ [3, 6, 16] and $K_b\approx 14k_{B}T$ [38] corresponding to MPPNs formed from [4, 7] MscS and diC14:0 lipids.

Figure 5

MPPN self-assembly phase diagram obtained from Eq. (4) as a function of protein radius ${\rho }_i$ and bilayer-protein contact angle $\alpha$. The white dashed line indicates the protein radius ${\rho }_i\approx 3.2\mathrm{nm}$ and $\alpha$-range associated with MscS [3, 6, 16]. We use the same labeling conventions as in Fig. 4 with the model parameter values $c\approx 7.8\times {10}^{-8}$ [4] and $K_b\approx 14k_{B}T$ [38] corresponding to MPPNs formed from [4, 7] MscS and diC14:0 lipids.