Analytical derivation of thermodynamic characteristics of lipid bilayer from a flexible string model

We introduce a flexible string model of the hydrocarbon chain and derive an analytical expression for the lateral pressure profile across the hydrophobic core of the membrane. The pressure profile influences the functioning of the embedded proteins and is difficult to measure experimentally. In our model the hydrocarbon chain is represented as a flexible string of finite thickness with a given bending rigidity. In the mean-field approximation we substitute the entropic repulsion between neighboring chains in a lipid membrane by an effective potential. The effective potential is determined self-consistently. The arbitrary chain conformation is expanded over eigenfunctions of the self-adjoint operator of the chain energy density. The lateral pressure distribution across the bilayer is calculated using the path integral technique. We found that the pressure profile is mainly formed by the sum of the partial contributions of a few discrete lowest-energy “eigenconformations.” The dependences on temperature and area per lipid of the lateral pressure produced by the hydrocarbon chains are found. We also calculated the chain contribution to the area compressibility modulus and the temperature coefficient of area expansion.

Figure 1

Model of lipid membrane in mean-field approximation.

Figure 2

Lateral pressure distribution in the hydrophobic core of the bilayer. The pressure arises from entropic repulsion between hydrocarbon chains in each monolayer. $z$ is the coordinate along the chain axis normalized by the chain length $L$ and spanning from $z=0$ at the head group to $z=L$ at the free chain end. The parameters for the lipid bilayer are as follows: chain length $L=15\mathrm{\AA }$, area per chain in all-trans conformation $A_0=20{\mathrm{\AA }}^2$, chain flexural rigidity $K_f\approx k_B{T}_{0}L∕3$, $T_0=300\mathrm{K}$, $P_{\mathrm{eff}}=100\mathrm{dyn}∕\mathrm{cm}$. Pressure is normalized by ${\Pi }_0=P_{\mathrm{eff}}∕3L$.

Figure 3

Hydrocarbon chain as a flexible string of finite thickness. $\mathbf{R}\left(z\right)$ is the vector characterizing the deviation of the center of the chain cross section from the $z$ axis, $\mid \mathbf{R}\left(z\right)\mid =\sqrt{R_x^2\left(z\right)+R_y^2\left(z\right)}$; $A_0$ is the “incompressible area” of the chain cross section; $A_1=\pi <{\mathbf{R}}^2>$ is the area swept by the centers of chain cross sections; $A$ is the average area per lipid chain in the bilayer.

Figure 4

The eigenfunctions $R_n\left(z\right)$ (normalized by $1∕\sqrt{L}$) of the self-adjoint operator $H_0$ for the given boundary conditions. Other parameters are as in Fig. 2.

Figure 5

Calculated lateral pressure $P_t$ produced by hydrocarbon chains as a function of area per chain at two temperatures $T_2$ (solid line) $>T_1$ (dashed line). Lateral pressure is normalized by $k_B{T}_0∕A_0$, area is normalized by $A_0$. Other parameters are as in Fig. 2.

Figure 6

Temperature dependence of equilibrium area per chain, $A$. Temperature is normalized by $T_0$, area is normalized by $A_0$.