Suppressing membrane height fluctuations leads to a membrane-mediated interaction among proteins

Membrane-induced interactions can play a significant role in the spatial distribution of membrane-bound proteins. We develop a model that combines a continuum description of lipid bilayers with a discrete particle model of proteins to probe the emerging structure of the combined membrane-protein system. Our model takes into account the membrane's elastic behavior, the steric repulsion between proteins, and the quenching of membrane shape fluctuations due to the presence of the proteins. We employ coupled Langevin equations to describe the dynamics of the system. We show that coupling to the membrane induces an attractive interaction among proteins, which may contribute to the clustering of proteins in biological membranes. We investigate the lateral protein diffusion and find that it is reduced due to transient fluctuations in membrane shape.

Figure 1

Snapshot of a simulation of the combined membrane-protein system. The gray surface shows the current membrane configuration; red dots illustrate protein positions. Simulation parameters are $P=Q=32, \sigma =0, \beta \kappa =10, \varepsilon =100k_{\mathrm{B}}T/{\sigma }_{\mathrm{p}}^2, N=100$.

Figure 2

Same snapshot as in Fig. 1, but showing in gray the potential energy surface $\varphi \left(\mathit{r}\right)$ instead of the membrane conformation. Proteins remain in the low-energy regions of this potential, whereas the membrane is free to fluctuate in protein-free regions.

Figure 3

(a) Dependence of membrane fluctuations on protein density. If no proteins are present the spectra are given by the free membrane result (25). As the protein density increases, the variance of membrane modes with small wave vectors is significantly reduced. Dashed lines represent numerical solutions of (26); points were obtained from simulations. (b) Dependence of membrane fluctuations on bending rigidity. The reduction of membrane fluctuations (relative to the free membrane result) is largest for flexible membranes that have low bending rigidity.

Figure 4

Effect of membrane rigidity on protein pair correlation functions at different densities. (a) At low densities ($N=2$) the RDF is essentially flat outside the protein diameter if the membrane is rigid, and develops a significant peak as the bending rigidity is reduced. (b) At intermediate densities ($N=100$) this effect is less pronounced. (c) At high densities ($N=500$) there is a peak in the pair correlation function, but it is due to packing and nearly independent of the membrane bending rigidity. (d) Free energy of the protein-protein interaction, Eq. (32), and membrane-mediated interaction, Eq. (30). Dashed lines are obtained from simulations; solid lines in panels (a) and (d) are obtained numerically as described in Appendix pp2.

Figure 5

Determination of the diffusion constant from mean-squared displacements. The time step $\mathrm{\Delta }t$ must be chosen sufficiently small so that observed MSDs do not depend on its value. The lag time $\tau$ must be large enough so that the dynamics reaches the diffusive regime, indicated by a constant value of $D$ as computed by (33). Uncertainties increase significantly with $\tau$ if the total length of the simulation data is fixed. Simulation parameters: $L=250\text{nm}, P=1, Q=0, N=1, D_0=5\times {10}^9{\mathrm{nm}}^2/\mathrm{s}, \beta \kappa =10, \sigma =0, \eta ={10}^{-29}\mathrm{J}\mathrm{s}/{\mathrm{nm}}^3, \varepsilon =100k_{\mathrm{B}}T/{\sigma }_{\mathrm{P}}^2$.

Figure 6

Renormalization of protein diffusion as a function of membrane bending rigidity $\kappa$ and solvent viscosity $\eta$. The ratio of actual to bare diffusion constant, $D/D_0$, varies strongly with system parameters both in the one-dimensional case [$P=1,Q=0$, panel (a)] and the most basic two-dimensional case [$P=1,Q=1$, panel (b)]. When plotted as a function of $D_0\eta /\kappa$ the data points collapse onto a single curve in both cases [panels (c) and (d), respectively].