Analytical calculation of the lipid bilayer bending modulus

Bending and Gaussian moduli of a homogenious single-component lipid bilayer are calculated analytically using microscopic model of the lipid hydrocarbon chains. The approach allows for thermodynamic averaging over different chains conformations. Each chain is modeled as a flexible string with finite bending rigidity and an incompressible cross-section area. The interchain steric repulsion is accounted for self-consistently determined single-chain confining parabolic potential. The model provides a simple analytical expression for the membrane bending modulus, which falls within a range of experimental values. An observed dependence of the modulus on hydrocarbon chain length is also reproduced. Correspondence between our microscopic model and the membrane theory of elasticity is established.

Figure 1

Hydrocarbon tails of lipid are modeled as a flexible string. Schematic representations.

Figure 2

Collisions with neighboring lipids are modeled by self-consistent confining potential. Potential is quadratic in the string deviation from axis $z$ (arrows size mimics local force strength).

Figure 3

Schematic representation of the lipid bilayer connected to a lipid reservoir. Area per lipid at the head group neutral surfaces of each monolayer is unchanged under bending provided there is a lipid reservoir. Condition Eqs. (26) and (27) for the monolayers are sketched in the figure.

Figure 4

Dependence of membrane bending—to stretch modulus ratio on the head-to-head thickness of fully saturated lipid bilayers at $T=323$ K. Stars are our model results; long dashes—polymer brush model results [51].

Figure 5

Dots: simulation data quoted from Ref. [32]—see lateral region for DPPC lipids in Fig. 9 there. Line: fit of toy model, Eq. (41). The fit leads to $n=1.6$, which results [see Eq. (45)] in $m=1/8.6$. This value is close to the one obtained with flexible strings model.