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 interactionsinhomogeneous 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 t vdW hgP P P P ... 0

hgP

vdWP

tP

total tension in a bilayer is zero

J.Israelachvili 1985; A.Ben-Shaul 1995

Lipid bilayer

Applications

0

L

Lact Edz)z()z(AE

activation energy of protein channel, Eact :

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, E0 – other contributions to the activation energy.

A(z)

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 thenof a flexible one

L

0

2

2

2

ft

L

0

2

t

))s(R(Vds

)s(Rd2

KdsE

))s(R(Vds

)s(Rd2KdsE

flexible chain

semi-flexible chain

Mapping on the quantum particle motion in imaginary time

Partition function as a path integral over chain conformations

}TEexp{)s(RDZ t

Flexible chain case:Burkhardt 1989, Vallade&Lajzerowicz 1981

)L,'R,R(G'Rd)L,R(Z 3

where Green’s function obeys:

Breidenich, Netz, Lipowsky 2000Semi-flexible chain case:Freed 1971, Gompper&Burkhardt 1989, Leibler et al. 1989

)'RR()L()L,'R,R(G)R(VK21

L R

wheredsRdu

)RR()uu()0,u,R;u,R(Z

0)L,u,R;u,R(Z)R(VK21u

L

0000

00u

f

R

Semi-flexible chain in harmonic potential

Alternative approach [S. Mukhin, 2004; S. Mukhin, S. Baoukina, 2005]:

L L2

2 i ift 2

0 0

K d R ˆE dz B(z)R (z) dz R (z)HR (z)2 dz

4

f4

K dH B(z)2 dz

possible with appropriate boundary conditions for at z=0,L.R(z)

i in n nHR (z) E R (z) i

n nR (z) C R (z)2

n nn

n

E CZ dC expT

n n

TF T lnE

22 n

n n

R (z)R (z) TE

0

z R(z)

tA B(z)(z)

A

4f n

n n n4

K R B(z)R E R2 z

Derivation of lateral pressure distribution

Lateral pressure profile can be found from the system of equations:

L

t t0

F(A)(z)dz PA

A – average area per chain

n

n n

EF 1TA A E

n n

0

E E B(z) dzA B(z) A z

2nn 0

E R (z)zB(z)

dz)z(dzA

)z(BE

)z(RTAF

tn n

2n

constAE

)z(RTn n

2n

A)z(B

E)z(RT)z(

n n

2n

t

general expression

0n n

2n

n n

2n

t

vALdzE

)z(RT

E)z(R

dAdBT)z(

)A(BB

constant density case variable density case

Approximate solution (constant density case)

Anzats:

)z(YE)z(YH nnn

where )z(Yn are eigen-functions:);z(YA2T)z(B

n

0

2

n

; where: )z(BHH 0

;z2

KH4

4

f0

and functions )z(Yn

are looked for in the form:

12021 RbRaY ;11010 RbRaY

unperturbed eigen-functions obey relations:

)z(RW)z(RH nnn0 unperturbed eigen values:

444

fn )1n4()4/)(L2/(KW 1n; 0W0

1ba 2

i

2

i 0baba 2211 ;

;

Approximate solution

1122

2/12

11

340011

3000

L

000

a;bb;a);1()1(b;a)II1(WE

)I1(E

dz)z(BL1

lateral pressure profile

8.1I4.0I17.0

4

3

Parameters:

cm/dyn150P;A20L;A40A t

002

mean-squared deviation

)z(R)z(RA2T)z(Y

A2T)z( 2

1

2

0

1

0

2

n

AE

)z(YT)z(R1

0n

2

n2

S.Mukhin 2004