Using microscopic molecular theory, we determine the bending and saddle-splay constants of three-component lipid bilayers. The membrane contains cholesterol, dipalmitoyl-phosphatidylcholine (DPPC) and dioleoylphosphatidylcholine (DOPC) and the predictions of the theory have been shown to qualitatively reproduce phase diagrams of giant unilamellar vesicles (GUVs) of the same three components. The bending and saddle-splay constants were calculated for the gel, liquid-ordered (lo) and liquid-disordered (ld) phases. By proper expansion of the free energy, the molecular theory enables us to determine the effects of the mode of membrane bending deformation on the value of the elastic constants for different phases. In particular, we refer to the ability of the molecules to arrange the composition between the two monolayers upon deformation. The bending and saddle-splay constants obtained from the free energy expansion can be expressed in terms of moments of the local lateral pressures and their derivatives, all evaluated for a symmetric planar bilayer. The effect of blocked vs. free exchange of lipids across the two monolayers on the values of the bending constant is as high as 50 kBT in the ld phase to as high as 200 kBT in the lo phase. These results show that one must strongly consider the mode of deformation in determining the mechanical properties of lipid bilayers. We discuss how the different contributions to the lateral pressures affect the values of the elastic constants, including the effects of the cholesterol concentration and temperature on the membrane elastic constants. We also calculate the equilibrium binding concentrations of lipid tail anchors as a function of membrane curvature by explicitly determining the chemical potential difference of species across a curved bilayer. Our results are in excellent agreement with recent experimental results.
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