Finite-size domains in membranes with active two-state inclusions

The distribution of inclusion-rich domains in membranes with active two-state inclusions is studied by simulations. Our study shows that typical size of inclusion-rich domains $\left(L\right)$ can be controlled by inclusion activities in several ways. When there is effective attraction between state-1 inclusions, we find: (i) Small domains with only several inclusions are observed for inclusions with time scales $(\sim {10}^{-3}\mathrm{s})$ and interaction energy $[\sim \mathcal{O}\left(k_{\mathrm{B}}T\right)]$ comparable to motor proteins. (ii) $L$ scales as $1∕3$ power of the lifetime of state-1 for a wide range of parameters. (iii) $L$ shows a switch-like dependence on state-2 lifetime $k_{12}^{-1}$. That is, $L$ depends weakly on $k_{12}$ when $k_{12}<k_{12}^{*}$ but increases rapidly with $k_{12}$ when $k_{12}>k_{12}^{*}$, the crossover $k_{12}^{*}$ occurs when the diffusion length of a typical state-2 inclusion within its lifetime is comparable to $L$. (iv) Inclusion-curvature coupling provides another length scale that competes with the effects of transition rates.

Figure 1

Schematics of a membrane with two-state inclusions.

Figure 2

Domain size distribution $P\left(M\right)$ for ${\varphi }_{\mathrm{inc}}=12.5\%$, $\Delta J=1.5k_{B}T$, $k_{12}={10}^2{\mathrm{s}}^{-1}$, $k_{21}=k_{12}∕32$, and $G=0$, 1, and 2. The peak at $M=1$ comes from the isolated inclusions in the inclusion-poor domain. The peak at greater $M$ provides the characteristic size of inclusion-rich domains. As $G$ increases, the characteristic size of inclusion-rich domains decreases and the peak of $P\left(M\right)$ becomes more significant due to inclusion-membrane coupling.

Figure 3

(Color online) Snapshots for steady state inclusion distribution (left, dark regions are occupied by inclusions) and membrane height (right) for ${\varphi }_{\mathrm{inc}}=12.5\%$, $\Delta J=1.5k_{B}T$, $k_{12}={10}^2{\mathrm{s}}^{-1}$, and $k_{21}=k_{12}∕32$. (a) $G=0$, (b) $G=2$. In order to faithfully present the morphology of the membrane, the unit length for $h$ and the unit length in the $xy$ plane are both chosen to be $a$. For nonzero $G$ membrane curvature close to the inclusion-rich domains increases and the typical size of inclusion-rich domain decreases.

Figure 4

Typical size of inclusion-rich domains in the steady state for ${\varphi }_{\mathrm{inc}}=12.5\%$, $\Delta J=1.5k_{B}T$, $k_{12}\Delta t={10}^{-3}$. $G=0$ (circles), $G=0.5$ (squares), $G=1.0$ (diamonds), and $G=2.0$ (triangles). The dashed line has slope $-1∕3$.

Figure 5

Typical size of inclusion-rich domains in the steady state for ${\varphi }_{\mathrm{inc}}=12.5\%$, $\Delta J=1.5k_{B}T$, $k_{21}\Delta t={10}^{-4}$. $G=0$ (circles), $G=0.5$ (squares), $G=1.0$ (diamonds), and $G=2.0$ (triangles). Data with greatest value of $k_{12}\Delta t$ are obtained from simulations on a $256\times 256$ lattice. The inset is a schematic of a domain with radius $L$. When $k_{12}>k_{12}^{*}$, state-2 inclusions in the inner circle (radius $\Delta L$) have small chance to leave the domain within its lifetime. $\Delta L=L-L_2$ increases as $k_{12}$ increases.