We found theoretically that competition between ~Kq^4 and ~Qq^2 terms in the Fourier transformed conformational energy of a single lipid chain, in combination with inter-chain entropic repulsion in the hydrophobic part of the lipid (bi)layer, may cause a crossover on the bilayer pressure-area isotherm P(A)~(A-A_0)^{-n}. The crossover manifests itself in the transition from n=5/3 to n=3. Our microscopic model represents a single lipid molecule as a worm-like chain with finite irreducible cross-section area A_0, flexural rigidity K and stretching modulus Q in a parabolic potential with self-consistent curvature B(A) formed by entropic interactions between hydrocarbon chains in the lipid layer. The crossover area per lipid A* obeys relation Q^2/(KB(A*))~1 . We predict a peculiar possibility to deduce effective elastic moduli K and Q of the individual hydrocarbon chain from the analysis of the isotherm possessing such crossover. Also calculated is crossover-related behavior of the area compressibility modulus K_a, equilibrium area per lipid A_t, and chain order parameter S.
Studying thermodynamics of the lipid bilayers that form biological membranes is of fundamental interest for understanding relation between membrane state and functioning of integral membrane proteins [1][2][3]. The latter are of vital importance for many processes in the living cells.
Experimental data in lipid membranes indicate presence of a cross-over in the pressure-area isotherms 0
A A α –∼ [4,5]. Formally, this means that exponent α changes substantially within some finite interval along the area axis A . A substantial amount of theoretical work has been devoted to description of the thermodynamic properties of lipid layers including pressure-area isotherms, chains order parameter as function of temperature, specific heat, etc. Theoretical approaches range from phenomenological Landau-de Gennes theory [6] to surface equations of states involving clustering [7][8][9][10] and raft formation [11]. Molecular dynamics [12] and Monte Carlo [13] simulations were done as well. Besides, the models were considered with phase transition due to change in the number of gauche conformations of the hydrocarbon chains [14][15][16][17][18][19][20],
as well as models focused on the role of the excluded volume interactions between the chains [21,22]. These factors were also combined in the form of the additive area dependent contributions to the surface pressure [20].
In the previous work [23] a theoretical method has been proposed of calculation of the thermodynamic characteristics of lipid bilayer starting from a “microscopic” model of a smectic array of semi-flexible strings of finite length with a given flexural rigidity, see Fig. 1. The string is an idealized model of the hydrocarbon chain. The entropic repulsion between the neighboring chains in lipid membrane is modeled with effective potential. This entropic potential is then found self-consistently, by minimizing free energy of the bilayer, that, in turn, is calculated using path-integral integration over possible conformations of the strings. As a result, the lateral pressure profile inside the lipid bilayer was derived analytically, together with the area compressibility modulus and the temperature coefficient of area expansion of the membrane. In [23] only bending energy of the strings and entropic repulsion were included in the conformational energy functional. In the Fourier transformed representation the former energy is proportional to
, where q is the wave-vector along the chain axis, and f K is chain flexural rigidity modulus. Then, resulting pressure-area isotherm of the lipid bilayer was derived in the form of a power law: 0 ( ) ~( )
Here the lateral pressure of the lipid hydrocarbon chains (tails) ( ) t P A is expressed as a function of the area per lipid A in the layer at a given temperature, with 0 A being the chain incompressible cross-section area. In the present work we added the stretching energy of the string to the energy functional [24]. In the Fourier transformed representation this energy is proportional to 2 Qq ∼ , where Q is chain stretching modulus. Hence, our new chain energy functional contains now the sum:
, where ( ) B A is self-consistently determined curvature of a parabolic effective entropic repulsive potential felt by a single chain due to surrounding chains in the lipid layer. The bending (flexural) energy dominates at large wave-vectors q , while the stretching energy dominates in the small q limit. The entropic repulsion term ( ) B A sets an upper limit for the wave-vectors q that are essential for thermodynamics. The entropic repulsion increases with a decrease of area per lipid in the layer, i.e.
As a result, the bending energy term
dominates, and we recover the previous pressure-area isotherm [23] with the exponent 5 / 3 α = . On the other hand, when the area per lipid increases, the entropic repulsion becomes weaker, and parameter ( ) B A becomes smaller. Hence, important q s ′ become smaller too and stretching energy term Also calculated are elastic moduli of the membrane and their dependences on temperature and on the microscopic elastic moduli of the individual chains constituting the lipid bilayer Figs. 4-6. Finally, we discuss how fitting of experimental isotherms of lipid bilayers with our theoretical isotherms may help to deduce elastic moduli K f and Q of individual lipid chains constituting the membrane.
The plan of the article is as follows. In the first part we formulate the physical model [24] of bilayer and review the path-integral method of summation over all conformations of idealized hydrocarbon chain [23]. In the second part we derive and solve analytically (in two different limits) a self-consistency equation for the curvature ( ) B A of the effective parabolic entropic potential in the layer. The pressure-area isotherms are then derived in the analytical form. In the third through fifth parts of the paper we present results of calculations of the thermodynamic and elastic characteristics of
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