БИОЛОГИЧЕСКИЕ МЕМБРАНЫ, 2015, том 32, № 1, с. 3-10

УДК 577.352


National University of Science and Technology "MISIS", Department for Theoretical Physics and Quantum Technologies,

Leninskiyprosp., 4, Moscow, 119049 Russia; *E-mail: kheyfboris@misis.ru Received 10.07.2014

Smaller area per molecule in a DPPC—cholesterol multicomponent monolayers than in pure DPPC suggests that DPPC lipids straighten when in contact with cholesterol. Using flexible strings model, that imitates en-tropic repulsion between lipid chains in the membrane, one can reproduce the DPPC—cholesterol area per molecule diagram at a low cholesterol concentration. Using an analytical interpolation we construct a pure cholesterol membrane, which allows us to calculate area per molecule in cholesterol "membrane" with small DPPC concentration. The last result suggests the area per lipid in a large cholesterol concentration DPPC membrane. The parameters found by fitting our model results to the experimental area—concentration diagram imply that cholesterol exerts greater lateral pressure in the membrane than DPPC.

Keywords: flexible strings model, lipid membrane, analytical calculation, path integral.

DOI: 10.7868/S0233475515010053


Cholesterol regulates fluidity of lipid membranes [1—3]. This ability of cholesterol is thought to be important in many biological processes including cell fusion [4], activity of the sodium pump [5], the functioning of the raft-embedded proteins [6], modulation of oxygen transport across red blood cell membranes [7], etc.

Measured area per lipid in DPPC—cholesterol mixtures is lower than concentration prorated sum of the corresponding areas in pure DPPC and cholesterol single component membranes [8, 9]. This suggests that DPPC occupies less space (straightens) when in contact with cholesterol. In this paper we show that a flexible strings model with a simple assumption about partial lateral pressures exerted by lipids of different kinds reproduces the area per lipid diagram of DPPC— cholesterol mixture. Analytical calculations within the framework of our flexible strings model shows interesting result. Namely, we found that if a DPPC lipid is inserted in a cholesterol-rich membrane, the change in the area per lipid is strongly nonlinear function of DPPC concentration. The reason is that area expansion of surrounding cholesterol molecules is relatively small, while area compression of the DPPC lipid is strong.

These effects are reflected in the theoretical results that are in correspondence with the experimental data presented in Fig. 1. The model parameters obtained from fitting the experimental data might be further

used to predict other thermodynamical properties of DPPC and cholesterol membranes, such as thermal coefficient of area expansion [11], pore formation pressure and critical stretch [12], bending modulus [13], etc. Ability to predict properties of multi-component lipid membrane, given the properties of its components, is an important issue in the physics of lipid membranes [14—16].

In the literature, the cholesterol-induced lipid phase is usually described as "liquid ordered", with more ordered hydrocarbon chains but only somewhat reduced rates of lateral lipid mobility [17]. Using a microscopic flexible strings model and allowing for different partial lateral pressures exerted by DPPC and cholesterol molecules inside a two-component lipid membrane, one can reproduce area per lipid curves. The values of the microscopic parameters entering the flextible strings model are extracted by comparison with experimental data and further used to calculate such thermodynamical quantities as thermal coefficient of area expansion [11], stable pore pressure, critical stretch [12], etc.

Flexible strings model is a microscopic mean-field model of single-component lipid membranes [11, 18]. Within this model lipid is kept in a membrane by a balance between repulsion of the hydrophobic tails, Pt, and attraction of the hydrophilic heads, y.

Attraction ofhydrophilic heads is due to hydropho-bicity of the hydrocarbon tails and Coulomb interac-




<D ft

50 -


------ - -


40 60

Cholesterol, %



Fig. 1. Data on area per molecule in DPPC-cholesterol mixtures. Solid line 1: averaged [9] results of computer simulations [24, 25, 30]. Dashed line 2: Data derived [8] from experiments [29] at surface lateral pressure 5 dyn/cm. Both curves suggest that DPPC area decreases when in contact with cholesterol.


tions of possibly charged heads [19]. We characterize it by a single integral parameter, y.

Repulsion of hydrocarbon tails is induced by excluded volume effect [20] and is calculated using other input parameters of the flexible strings model: tail incompressible area, An, tail stiffness, Kf, and tail length.

Consider two types of lipids which have different y in their own membranes. An and Kf might also be different. Let us insert lipid of one kind in the membrane consisting of lipids of the other kind. Then a mismatch of y might be interpreted as a pressure that lipid experiences from the rest of the membrane. Hence, when lipids have different y, there would be additional (positive or negative) pressure around the alien lipid. One might model that situation using a simple exponential with a characteristic length of R0 for solvent lipids and a given value (determined generally by the chemical structure of the components) of y solute of the solute lipid. The perturbation of y might be due to chemical difference of polar heads, or due to different Coulomb charges of the heads.

It turns out that choosing ysolute « yChol and a small enough R0, of the order of average radius of the area occupied by the molecule, allows one to reproduce dependence of the area per molecule on concentrations. The overall approach is limited to small concentrations only.

CALCULATIONS DPPC lipids as flexible strings

We modeled DPPC as flexible strings [11]. Here we reproduce the basics. For more details we refer reader to the original publications [11, 18].

DPPC lipids are modeled as flexible strings with finite thickness and corresponding incompressible area An (see Fig. 2) and finite bending rigidity Kf of the hydrocarbon chain. Energy functional of the hydrocarbon tail is written in terms of deviations of the string from the straight line along the z-axis, R(z):

L T 2

^ = f + ^ f^] + BR! dz. (1)

' J [ 2 2 Uz2) 2 J

Here L is thickness of the hydrophobic part of the membrane monolayer, p is a line density. First term here is kinetic energy, second is bending energy of the tail, and the last term models interaction of lipid with neighbors using a mean field approximation. Its parameter B is to be determined self-consistently.

Using boundary conditions for the string, it is possible to rewrite energy functional using operator of chain ("string") conformational energy. Its eigen functions might be considered as elementary oscillation modes of the string. Writing down partition function of the string as a product of the elementary modes and equating its derivative over B to the average area swept by the string allows one, in turn, to determine interaction parameter B self-consistently.

Chain's free energy consists of oscillation of the tails and surface energy:

FT = Ft +jA,


where A is a mean area swept by the chain. Requiring free energy of the chain be at minimum as a function of A one derives an equilibrium condition:

dF + y = 0. dA


Noting that lateral pressure of the chains is Pt = = -dFt /dA, one finds that equilibrium condition actually requires that repulsion of the tails must be balanced by the attraction of the heads.

The derivation finally leads us to



An 3v1/3Va (( -1)


= Y.



Here v = — n is a dimensionless parameter. This nkBTL

equation allows one to determine dimensionless mean area swept by the chain, a = A/An.

It is interesting to note that only two dimensionless

parameters: v and g = AnY define the average area


swept by the chain. The first one characterizes bending energy of the lipid chain conformation (flexible "string" in our model), and the last one is "surface energy" per lipid. Both measured in kBT.


Fig. 2. Hydrocarbon tail as a flexible string. A is a mean area swept by the chain. An is incompressible area.

Equation (6) provides an average area swept by the rigid rod (due to lateral vibrations of the rod as a whole):

1 + |1 + 4 k^T




Note that since bending energy is infinite, only one parameter defines mean area swept by the hard rod: di-

mensionless "surface energy" g = .


Cholesterol molecule as a rigid rod

We modeled cholesterol molecules as rigid rods [12]: bending rigidity modulus of cholesterol hydrocarbon chain is taken infinite: Kf = to . In this case bended conformations of the lipid chains are rare, and so bending does not contribute to the lowest energy of the chain:

Et =

pR2 + BR2



Equilibrium condition (3) holds, and Eq. (4) takes the form of

k„T 1

An a




This result might be obtained from flexible string energy functional Eq. (1) if one takes into account that Kf is large, then v is large (see its definition below Eq. (4)). Then, different approximation for a self-consistency equation (not shown in this short overview) must be taken, this leads directly to Eq. (6).

Area per lipid

In order to describe the straightening effect we assume that y Choi > YDPPC. For DPPC there is an estimate of surface tension at the hydrophilic interface made on the basis ofundulation analysis [21]: y = 50 erg/cm2. Generally, y might be obtained with molecular dynamics simulation by taking integral over the lateral pressure profile near the heads [22].

We choose our parameters in order to reproduce area per lipid for pure DPPC [23]. For our analytical interpolation tow

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