Hybrid Elastic and Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins

We introduce a simulation strategy to consistently couple continuum biomembrane dynamics to the motion of discrete biological macromolecules residing within or on the membrane. The methodology is used to study the diffusion of integral membrane proteins that impart a curvature on the bilayer surrounding them. Such proteins exhibit a substantial reduction in diffusion coefficient relative to “flat” proteins; this effect is explained by elementary hydrodynamic considerations.

Figure 1

An appropriately shaped protein will tend to distort the local shape of the bilayer into a bent configuration. In our simulations, the protein’s effective size is defined via the envelope function in Eq. (5) (displayed next to the cartoon, top). The distortion is most readily seen by minimizing $\mathcal{H}$ for the composite protein-bilayer system (middle); however, the protein’s influence is strong enough to maintain visibly apparent perturbations, even in the presence of fluctuations (bottom). See Table  for parameters.

Figure 2

Projected diffusion coefficients for protein motion on an elastic membrane. In each panel, one of the three parameters $K_p$ (protein bending modulus), $C_p$ (protein spontaneous curvature), and $\eta$ (medium viscosity) is varied as indicated while holding the remaining physical properties at the default values indicated in Table . Circles indicate the results of full simulations including thermal motion of both the bilayer and the protein. The triangles indicate results obtained by turning off all bilayer shape fluctuations (i.e., setting $T=0$ for membrane undulation modes), and the solid squares indicate the results of the adiabatic theory discussed in the text. The insets show simulation results obtained within the asymmetric coupling approximation. Error bars are approximately the size of the symbols. Diffusion coefficients are calculated as $D=\overline{[\mathbf{r}(t)-\mathbf{r}(0){]}^2}/(4t)$. The simulations run for $\sim 5\times {10}^7$ time steps of size $\Delta t\sim 0.01\mathrm{ns}$, ensuring both numerical accuracy and statistically reliable results. The protein’s root-mean-square displacements over the course of the simulations are approximately 100 nm (i.e., 20 times the protein’s radius).