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.