Figure 5.

(A) Ratio ${\displaystyle K_{2\mathrm{D}}/K_{3\mathrm{D}}}$ of the binding constants of lipid-anchored and soluble receptors and ligands and (B) on-rate constant ${\displaystyle k_{\mathrm{o}\mathrm{n}}}$ of lipid-anchored receptors and ligands vs. average membrane separation ${\displaystyle \overline{l}}$ of 2 membranes with area 14 × 14 nm² and a single lipid-anchored receptor and ligand in our coarse-grained molecular model. The relative membrane roughness is determined by the membrane area in this system and attains the value ${\displaystyle {\xi }_{\perp }=0.54\pm 0.01}$ nm. The data points result from MD simulations. The full lines in (A) result from a fit of our general theoretical results for ${\displaystyle K_{2\mathrm{D}}/K_{3\mathrm{D}}}$ from Eqs. (1), (7), and (8) with fit parameters ${\displaystyle c_{2\mathrm{D}}=480\pm 20}$ nm², ${\displaystyle L_0=10.64\pm 0.02}$ nm, and ${\displaystyle k_{RL}=7.2\pm 0.7}$ ${\displaystyle k_{B}T/\mathrm{n}{\mathrm{m}}^2}$ for the anchoring strength ${\displaystyle k_a\simeq 2.5}$ ${\displaystyle k_{B}T}$ of our lipid-anchored receptors and ligands. The full lines in (b) result from a fit of our general theoretical results for ${\displaystyle k_{\mathrm{o}\mathrm{n}}}$ from Eqs. (9), (10), and (2) with fit parameters ${\displaystyle c_{on}=77\pm 4}$ ${\displaystyle \mu {\mathrm{m}}^2/\mathrm{s}}$, ${\displaystyle L_{\mathrm{T}\mathrm{S}}=10.63\pm 0.02}$ nm, and ${\displaystyle k_{\mathrm{T}\mathrm{S}}=1.5\pm 0.2}$ ${\displaystyle k_{B}T/\mathrm{n}{\mathrm{m}}^2}$. The dashed lines represent fits to Eqs. (17) and (18) obtained for the classical Gaussian theory with fit parameters (a) ${\displaystyle K_{2\mathrm{D}}^{\mathrm{m}\mathrm{a}\mathrm{x}}=146\pm 9}$ nm², ${\displaystyle l_K=9.36\pm 0.06}$ nm, and ${\displaystyle {\xi }_K=1.08\pm 0.04}$ nm and (b) ${\displaystyle k_{\mathrm{o}\mathrm{n}}^{\mathrm{m}\mathrm{a}\mathrm{x}}=42.1\pm 1.4}$ ${\displaystyle \mu {\mathrm{m}}^2/\mathrm{s}}$, ${\displaystyle l_k=9.35\pm 0.05}$ nm, and ${\displaystyle {\xi }_k=1.41\pm 0.05}$ nm.