Curvature sorting of proteins on a cylindrical lipid membrane tether connected to a reservoir

Membrane curvature of a biological cell is actively involved in various fundamental cell biological functions. It has been discovered that membrane curvature and binding of peripheral membrane proteins follow a symbiotic relationship. The exact mechanism behind this interplay of protein binding and membrane curvature has not yet been properly understood. To improve understanding of the mechanism, we study curvature sorting of proteins in a model system consisting of a tether pulled from a giant unilamellar vesicle using mechanical-thermodynamic models. The concentration of proteins bound to the membrane changes significantly due to curvature. This has also been observed in experiments by other researchers. We also find that there is a phase transition based on protein concentration and we discuss the coexistence of phases and stability of solutions. Furthermore, when sorting is favorable, the increase in protein concentration stabilizes the tether in the sense that less pulling force is required to maintain the tether. A similar mechanism may be in place, when motor proteins pull tethers from donor membranes.

Figure 1

Schematic of a catenoid along with the definition of geometric variables.

Figure 2

Numerical solution of the shape equations for Van der Waals mixing energy with $a=3$, $b=1$, $\Sigma =20$, $L_{\mathrm{tube}}=12,$ and $C_0$ as shown. (a) The shape (thin-solid, dashed, and dash-dot lines) and the variation protein concentration (thick-solid, dashed, and dash-dot lines) along the length for ${\theta }_{\mathrm{ves}}=0.5$. The catenoid shapes for different values of $C_0$ do not differ significantly from each other. The variations (b) $\theta \left(S_{\mathrm{end}}\right)$ with ${\theta }_{\mathrm{ves}}$; (c) normalized $\theta \left(S_{\mathrm{end}}\right)$ with ${\theta }_{\mathrm{ves}}$ (an enlarged view for small ${\theta }_{\mathrm{ves}}$ has been shown in inset); and (d) pulling force $f$ with ${\theta }_{\mathrm{ves}}$.

Figure 3

Numerical solution of the shape equations for Bragg-Williams mixing energy with $\chi =1.5$, $b=1$, $\Sigma =20$, $L_{\mathrm{tube}}=12,$ and $C_0$ as shown. (a) The shape (thin-solid, dashed, and dash-dot lines) and the variation protein concentration (thick-solid, dashed, and dash-dot lines) along the length for ${\theta }_{\mathrm{ves}}=0.5$. The catenoid shapes for different values of $C_0$ do not differ significantly from each other. The variations (b) $\theta \left(S_{\mathrm{end}}\right)$ with ${\theta }_{\mathrm{ves}}$; (c) normalized $\theta \left(S_{\mathrm{end}}\right)$ with ${\theta }_{\mathrm{ves}}$ (an enlarged view for small ${\theta }_{\mathrm{ves}}$ has been shown in inset); and (d) pulling force $f$ with ${\theta }_{\mathrm{ves}}$.

Figure 4

Comparison of some quantities obtained from numerical solution of the catenoid shape equations and the algebraic equations for the cylinder with $b=1$ and $L_{\mathrm{tube}}=12$. (a) Tube protein concentration and (b) pulling force vs ${\theta }_{\mathrm{ves}}$ for the Van der Waals gas model for $a=3$. (c) Tube protein concentration and (b) pulling force vs ${\theta }_{\mathrm{ves}}$ for the Bragg-Williams model for $\chi =1.5$. Other parameter values are as mentioned in the plots.

Figure 5

Normalized tube protein concentration vs $1/\sqrt{\Sigma }$ or the initial tube radius for the Van der Waals (left) and Bragg-Williams (right) models. For small initial tube radius we observe a large increase in protein density in the tube.

Figure 6

(a) and (b): Variation of $\sqrt{\Sigma }$ and pulling force $f$, respectively, with protein coverage fraction for the Van der Waals interaction parameters $a$ above and below the critical value. (c) Pulling force with $\sqrt{\Sigma }$ for $a$ above $a_{\mathrm{crit}}$. A Gibbs loop is seen and the self-intersection corresponds to the values of $f$ and $\Sigma$ at coexistence. (d) The binodal (thick solid) and spinodal (dash-dot) curves in $\theta -\sqrt{\Sigma }$ plane. The points on the binodal are obtained by finding the outermost intersections (marked by open circles) of the vertical line from the coexistence point in (c) with the $\theta$ vs $\sqrt{\Sigma }$ (thin solid) curve. The points on the spinodal corresponds to the turning points (filled circles) of the $\theta$ vs $\sqrt{\Sigma }$ curve.

Figure 7

Coexistence curve for Van der Waals model for different ${\theta }_{\mathrm{ves}}$ for $b=1$. Ranges of values of $\theta$ and $\Sigma$ become narrower with increasing ${\theta }_{\mathrm{ves}}$.

Figure 8

Comparison of model prediction with experimental results. Best fit curves to experimental data of Figs. (a) 2E and (b) 2F of Zhu et al. [33] for the Van der Waals and the Bragg-Williams models. Fitted parameter values are shown in Table .

Figure 9

Comparison of model prediction with experimental results. Best fit curves to experimental data of Figs. (a) 2A, (b) 2B, and (c) 2C of Sorre et al. [29], respectively, for the Van der Waals and the Bragg-Williams models. Fitted parameter values are shown in Table .