БИОЛОГИЧЕСКИЕ МЕМБРАНЫ, 2010, том 27, № 4, с. 366-376
УДК 577.352.2
ANALYTICAL DERIVATION OF THERMODYNAMIC PROPERTIES OF BILAYER MEMBRANE WITH INTERDIGITATION © 2010 S. I. Mukhin, B. B. Kheyfets*
Department of Theoretical Physics and Quantum Calculations,State University of Technology and Science "MISA", Moscow,
119049 Russia; e-mail: sergeimoscow@online.ru *Physical Chemistry Department, State University of Technology and Science "MISA", Moscow, 119049 Russia
Received 16.10.2009
We consider a model of bilayer lipid membrane with interdigitation, in which the lipid tails of the opposite monolayers interpenetrate. The interdigitation is modeled by linking tails of the hydrophobic chains in the opposite monolayers within bilayer as a first approximation. This model corresponds to the types of interdigitation that are not related with the areal "hydrophobic" dilation of the membrane. A number of essential thermodynamical characteristics are calculated analytically and compared with the ones of a regular bilayer membrane without interdigitation. Important difference between lateral pressure profiles at the layers interface for linked and regular bilayer models is found. In the linked case, the lateral pressure mid-plane peak disappears, while the entropy decreases and the free energy per chain increases. Within our model we found that in case of elongation of the chains inside a nucleus of, e.g., liquid-condensed phase, homogeneous interdigitation would be more costly for the membrane's free energy than energy of the hydrophobic mismatch between the elongated chains and the liquid-expanded surrounding. Nonetheless, an inhomogeneous interdigitation along the nucleus boundary may occur inside a "belt" of a width that varies approximately with the hydrophobic mismatch amplitude.
Key words: bilayer membrane, interdigitation, analytical derivation, lateral pressure profile, hydrophobic mismatch.
1. INTRODUCTION
Studying mechanisms of changes in the structure of cell membrane under the adsorption of small am-phiphilic molecules (alcohol, anesthetics, etc.) is of fundamental interest, as well as is important for understanding of the functioning of the cell membranes and embedded proteins [1]. One of the drastic changes of the structure is membrane transition into the interdig-itated phase [2]. While in a regular membrane the thickness of the hydrophobic part of a bilayer is approximately twice the length of the hydrophobic tails of the phospholipid, an interdigitation may reduce the hydrophobic thickness to the sum of the length of the lipid tail and of the small amphiphilic molecule. Understanding possible consequences of the interdigita-tion for the lipid membrane properties is important for, e.g., prediction of the effects of anesthetics on the functioning of the ion channels embedded in the membrane [3, 4].
As a first approximation to the lipid bilayer membrane with interdigitation, in which lipid tails from the opposite monolayers interpenetrate, we consider a model with pairwise linked tails ofthe lipids belonging to the opposite monolayers within a single bilayer (Fig. 1a, b). Our model does not allow for a lateral area dilation of the membrane that may follow the interdigitation [1] but it bears an important property of the interdigitated bilay-
er in the form of constrained meandering freedom of the chains ends in the vicinity of the monolayer interface. Besides, there are types of interdigitation that are not related with the areal dilation of the membrane (Fig. 1c, d). We found important consequences of this constriction (linking chains): the entropy of the bilayer decreases, the free energy increases, and the lateral pressure profile nt(z) changes drastically.
The two distinct nt(z) curves (Fig. 2a) can be understood by comparing orientational fluctuations of the hydrocarbon segments of the semi-flexible lipid chains in the interdigitated and non-interdigitated lipid bilayers. These fluctuations can be characterized by an orientational order parameter S(z) (Fig. 2b, z is coordinate measuring depth inside bilayer), calculated using our model. The fluctuations reach their maximum at the monolayer interface inside a non-interdig-itated bilayer, because the chains ends are free there. Hence, the order parameter drops at z = L (Fig. 2b, dashed curve). Simultaneously, a maximum of the en-tropic lateral pressure occurs at z = L (Fig. 2a, dashed curve). Distinctly, in the interdigitated bilayer fluctuations are significantly suppressed at the monolayer interface due to restriction of orientational freedom of the central segments by their peripheral neighbors. Hence, S(z) does not drop at z = L (Fig. 2b, solid curve). As a consequence, there is no maximum at z = L
A
oo#oo|
• #
Fig. 1. a, Membrane with interdigitation. b, Our model of membrane with interdigitation. c, Left one is so-called partial interdigitation and the right one is so-called mixed interdigitation (see [22] for details). d, Interdigitation by alcohols (see [1] for details).
0.8 0.6 0.4 0.2
-with interdigitation
^ —no interdigitation-
0.5
1.0 z/L
1.5
2.0
S(z)
1.0
0.8 0.6 0.4
J
-with interdigitation
----no interdigitation
0.5
1.0 z/L
1.5
2.0
Fig. 2. a, Lateral pressure distribution in the hydrophobic core of the bilayer with (solid line) and without interdigitation. z is coordinate along the chain axis normalized by the monolayer thickness L and spanning from one head group (z = 0) to another (z = 2L). n° = (14/3) x 108 dyn/cm2. The parameters for the lipid bilayer are as follows: monolayer thickness L = 15 A, area per chain = 20 A2, chain flexural rigidity K = kBTL/3, temperature T = 300 K, total monolayer pressure P = 70 dyn/cm. b, Order parameter in cases with interdigitation (see Fig. 1) and no interdigitaion (see Fig. 3a).
vw
d
c
0
0
in the nt(z) dependence (Fig. 2a, solid line) for the in-terdigitated bilayer case.
We present our analytical results derived in closed form for thermodynamical properties of a membrane with linked chains in the weak interdigitation limit: i.e., thickness of the hydrophobic part of a bilayer is comparable with twice the length of a hydrophobic chain. In this limit we use more complete version of the energy functional entering the membrane partition function than developed earlier [5]: besides the bending energy of a chain conformation, we included kinetic energy of the lipid chain. We prove that this makes the path integral representation of the free energy of the chains uniquely normalizable.
The plan of the article is as follows. In Section 2 we introduce a microscopic model of a membrane with interdigitation and calculate the membrane free energy using path-integral summation over the chains conformations. The inter-chain entropic interactions are
treated in the mean-field approximation. Several thermodynamic moduli characterizing the interdigitated bilayer are derived as well and compared with non-in-terdigitated case. An increment of the free energy per chain due to interdigitation is calculated. In Sections 3 and 4 we calculate analytically the lateral pressure distribution (profile) across the hydrophobic core of the lipid bilayer and make comparison between the cases with- and without interdigitation. Also calculated is chain order parameter that characterizes correlations between the orientations of the chain segments and clearly demonstrates an increase of the orientation order in the interdigitated case as compared with non-interdigitated bilayer. In Appendix B we discuss possibility of inhomogeneous interdigitation along the boundary of the transmembrane liquid-condensed domain (raft) embedded in the liquid expanded surrounding in the bilayer membrane. We evaluate energetically favorable configuration of such a raft allowing
single chain
+ U
eff
bilayer
г
Fig. 3. a, Model of lipid membrane in the mean-field approximation: we substitute interaction between neighboring chains by an effective quadratic potential. b, Hydrocarbon chain as a flexible string of finite thickness. R(z) is the vector characterizing the de-
chain cross section;
A = n(R2) is the area swept by the centers of chain cross sections; A is the average area per lipid chain in the
bilayer.
I 2 2
viation of the center of the chain cross section from the z axis, |R(z)| = ,jRx(z) + Ry(z) ; Aq is the "incompressible area" of the
for the trade between hydrophobic mismatch and in-terdigitation-induced free energy increase.
2. MICROSCOPIC MODEL OF INTERDIGITATED BILAYER
The interdigitated lipid bilayer membrane is modeled by linking pairwise the tails of the chains belonging to the opposite monolayers. Hence, a couple of linked chains is substituted by a single semi-flexible string of length ~2L, where L is the monolayer thickness (Fig. 3a). Correspondingly, conformations of the string as a "transmembrane" object obey combined boundary conditions at the opposite head group regions of bilayer with coordinates г = 0 and г = 2L respectively, as is described below in detail.
With bending (flexural) rigidity Kf and with the mean-field approximation accounting for entropic repulsion between neighboring couples of pairwise linked chains (see Fig. 3a), the energy functional of a single string, E, has the form:
2L
E = J
0
P R (г ) + К(e2R(z)
. 2 2 ( dz2
+ 2R2 (z) \dz. (1)
Here harmonic potential Uf = BR2/2, with self-con-sistently defined rigidity B, describes entropic repulsion between the strings, z is coordinate along the string axis, and R(z) is vector in the {x, y} plane characterizing deviation of the string from the straight line,
R2 = R^ + R^ (see Fig. 3b). The choice of harmonic potential is justified since we assume finite "softness" of the effective "cage" created by the neighboring lipid
chains in the limit of small chain deviations. A harmonic potential was considered in earlier work [8] for a semi-flexible polymer confined along its axis. The first term in Eq. (1) represent
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