Moscow State Institute for Steel and Alloys Department of Theoretical Physics Analytical derivation of thermodynamic characteristics of lipid bilayer Sergei Mukhin Svetlana Baoukina Benasque, Spain, 2005

Lateral pressure in lipid bilayer * amphiphilic nature * elongated structure * spatial separation of different interactions inhomogeneous pressure profile hydrocarbon chains: * rotational isomers * N segments flexibility large conformational entropy in a bilayer: * collisions between chains * excluded volume effect entropic repulsion lateral pressure distribution in the hydrophobic core total tension in a bilayer is zero J.Israelachvili 1985; A.Ben-Shaul 1995

Lipid bilayer

Applications activation energy of protein channel, E act :  is difficult to measure experimentally due to complex intermolecular interactions and nanometer scale of membrane thickness;  effects the functioning of membrane proteins (when cross-section area varies with depth under protein conformational transition) [R.S. Cantor, Chem. Phys. Lipids. 101, 45 (1999) ] - change of the channel cross-section area at position z under channel activation, E 0 – other contributions to the activation energy. Lateral pressure profile in a lipid membrane:

Fluctuating chain in external potential: overview Free energy of fluctuating chain in the external potential Mapping of the chains statistics on the quantum particle motion in the imaginary time Why mapping of a semi-flexible chain is more involved then of a flexible one flexible chain semi-flexible chain

Mapping on the quantum particle motion in imaginary time Partition function as a path integral over chain conformations Flexible chain case: Burkhardt 1989, Vallade&Lajzerowicz 1981 where Green’s function obeys: Breidenich, Netz, Lipowsky 2000 Semi-flexible chain case: Freed 1971, Gompper&Burkhardt 1989, Leibler et al where

Semi-flexible chain in harmonic potential Alternative approach [S. Mukhin, 2004; S. Mukhin, S. Baoukina, 2005]: possible with appropriate boundary conditions for at z=0,L. 0 z R(z)

Derivation of lateral pressure distribution Lateral pressure profile can be found from the system of equations: A – average area per chain general expression constant density case variable density case

Approximate solution (constant density case) Anzats: whereare eigen-functions: ;where: and functions are looked for in the form: ; unperturbed eigen-functions obey relations: unperturbed eigen values: ; ;

Approximate solution lateral pressure profile Parameters: mean-squared deviation

S.Mukhin 2004