University of Ljubljana Faculty of Matematics and Physics Department of Physics Jadranska 19, Ljubljana

Seminar II

Phase Transitions in Lipid Membranes

April 2009 Author: Jernej Mazej

Advisors: doc. dr. Janez Štrancar dr. Zoran Arsov Abstract: The seminar treats phase transitions in lipid membranes, with the focus on the main lipid bilayer phase

transition. It reveals the microscopic picture and reviews the key factors leading to the presented

phenomena. A general discussion on models is made and some concrete examples are presented in

more detail. Finally, experimental approaches to study the phase behavior of lipid membranes are

discussed and the biological significance is portrayed.

2

CONTENTS

1. LIPID MEMBRANES 3

2. MICROSCOPIC PICTURE AND THE ORIGIN OF PHASES 5

3. THEORETICAL MODELS 7

- GENERAL REMARKS ON MODELS AND FACTORS

INVOLVED IN MLBPT 7

- SOME EXAMPLES OF MODELS 8

4. EXPERIMENTS 13

- CALORIMETRY AND DILATOMETRY 13

- X-RAY MEASUREMENTS 13

- EPR SPECTROSCOPY 14

- OTHER METHODS 15

5. BIOLOGICAL SIGNIFICANCE AND APPLICATIONS 16

6. CONCLUSION 17

7. REFERENCES 18

3

LIPID MEMBRANES

Lipids represent a very vast and heterogeneous group of fat-soluble organic molecules. They are

amphiphilic molecules consisting of a polar »headgroup« and one or more non-polar fatty acyl

chains which leads to the well known phenomena of self-assembly in water due to hydrophobic force

even at fairly low concentrations (for example, two-chain lipids can form vesicles even in 10-11 mol/L

solutions) [*1]. Self-assembly of lipid molecules can lead to different structures that are determined by

the interplay of the hydrophobic effect, various intra- and intermolecular interactions, structural

constraints and external conditions, e.g. water content. Such lyotropic structures are micelles, cubic

phases, inverted hexagonal phase and lamellar phases such as lipid bilayers [Fig. 1]. Biologically most

relevant is the lipid bilayer, which represents the main constitutive part of biological membranes in

cells. In this seminar, we shall limit ourselves to the bilayer case.

Lipid bilayers are membranes consisting of two monolayers of lipids organized mainly in planar

configuration. In the two monolayers of a bilayer lipids are oriented in the opposite directions with the

hydrophilic heads facing the outer water medium and »shielding« the hydrophobic tails from coming

into contact with water. Also for that reason, the interstitial region should not be filled with water

molecules, so both layers are tightly close to each other (the thickness of a bilayer is typically ~5 nm

for lipids with about 10 or 20 carbon atoms in the acyl chain). Moreover, the contact area becomes

even smaller if the boundaries of the bilayer join to each other resulting in an enclosed spherical form

of the bilayer – a liposome or vesicle.

Besides the hydrophobic one, there are also other interactions present [*2]: hydrogen bonds, van der

Waals forces and Coulomb interactions (for example between charged lipid molecules or even

between dipoles, since the lipid molecules exist mostly as zwitterions in a large range of pH). Some of

those interactions are long-ranged some of short range, which can lead to a complex phase behavior

and to phase coexistence, even in single lipid systems. To illustrate this expectation, recall an

analogous situation in ferromagnetic systems where the competing short-range exchange interaction

and long-range dipolar interaction give rise to the formation of coexisting domains.

Interestingly, some dilatometric and calorimetric measurements on lipid bilayers [*3] have revealed

jumps in molecular volume (volume per lipid) and sudden changes of the heat capacity while varying

the temperature [Fig. 2]. The existence of latent heat well indicates a phase transition.

Figure 1: Examples of lyotropic phase structures: a) a micelle (one half of); b) inverted hexagonal phase; c) lipid bilayer. In addition to the structures mentioned in the text, some other forms (e.g., inverted rhombohedric) were also detected in a few lipid systems at low hydrations. [*1]

4

Furthermore, a transition of a sharp Bragg peak into diffuse one near Tm in small angle X-ray

diffraction experiments on lipid membranes [*4] also demonstrate the structure passing from a more

ordered phase into a less ordered one.

Regarding the continuity of the first derivatives of the free enthalpy (e.g., entropy and volume), a

common division of phase transitions is to first order transitions (with latent heat) and to second order

transitions with continuous changes in first order derivatives of the free enthalpy but jumps in the

second order derivatives. Materials can exhibit both types of transitions in different regions of the

phase space, and at the so called critical values of quantities the discontinuity of the transition can

vanish. Another possible formalism for describing phase transitions is (according to Landau) to

introduce an order parameter, a physical quantity that is exactly zero for the disordered phase and

different from 0 for various ordered ones. Characteristic dependencies of the order parameter for both

types of transitions are depicted in [Fig. 3]. As we shall see from the microscopic structure and the

similarity between lipid bilayers and smectic phases of liquid crystals, an order parameter analogous to

the orientational one of LC systems is also adequate to describe order in lipid bilayers.

Figure 3: a) Typical temperature dependence of the order parameter of the first-order nematic-to-isotropic-phase transition in liquid crystals. The orientational order parameter S is traditionally defined as ½ <3cos2θ – 1>, where θ is the angle between each molecule's long axis and the director. The brackets denote an average over the whole sample. [*5]

b) Temperature dependence of magnetization (M) as an example of an order parameter of the second-order phase transition from ferromagnetic to paramagnetic (FM-PM) phase of a ferromagnetic system. In contrast with first-order transitions, the order parameter diminishes to zero continuously. [*6 ]

Figure 2: Molecular volume (circles) and heat capacity (solid line) vs. temperature for DPPC bilayers in excess water. [*3]

5

MICROSCOPIC PICTURE AND THE ORIGIN OF PHASES

For a better understanding of the origin of different phases, we first consider a simple microscopic

picture of a lipid molecule in a lipid bilayer.

A typical lipid, e.g. DPPC (Dipalmitoylphosphatidylcholine; Fig. 4, a-b), is chemically the glycerol-

ester of two palmitic acid tails and one phosphorilated choline headgroup. In an even more simplified

view, we can regard it as a simple amphiphile presented in Fig. 4c.

Due to this structural properties there are several different possibilities for order/disorder. Firstly –

disregarding the structure of the molecules – positional order/disorder is to be taken into account.

Secondly, because of the high asymmetry of lipid molecules, one could also achieve orientational or

headgroup order/disorder by taking molecules of the same shape and rotating them along their long

axes. As one cannot expect that all the molecules are perpendicular to the bilayer plane, there is also

tilt order/disorder. Namely, the molecules can lower their free energy by tilting or shifting

neighbouring chains and decreasing an area per molecule. Furthermore, by fixing the headgroup

orientation, one does not exhaust the possibility for chain orientation order/disorder and, finally,

because of many internal conformational degrees of freedom within a hydrocarbon chain,

conformational or rotameric order/disorder should be of great importance [Fig. 6]. Regarding the

variety of order/disorder possibilities, the lipid bilayers in water should and actually do undergo a

plenty of phase transitions when varying the temperature [Fig. 2]. The majority of phases

corresponding to different order/disorder types has been at least indirectly demonstrated

experimentally and properly named [Fig. 5].

Figure 5: Sketched examples of some characteristic lipid bilayer phases: Lβ – (untilted) gel state, Lβ' – tilted gel state, Pβ' – ripple phase; Lα – liquid crystalline phase. [*1]

Figure 4: a) structural formula and b) 3D model of a DPPC molecule; c) a more simplified picture of few molecules in one half of a bilayer.

6

In our seminar we will treat in more detail only the transition, which occurs at the highest temperature

and represents the largest enthalpic change. This is the so called main lipid bilayer phase transition

(or MLBPT) and corresponds characteristically (though not solely) to the activation of the rotameric

degrees of freedom.

Figure 6: Molecular dynamics simulations of a PC bilayer in gel phase (left) and liquid crystalline phase (right). Note the considerable portion of lipid molecules in a straight (i.e., »all-trans«) conformation in the gel phase, while in the fluid phase chains exhibit kinks (»gauche defects«) which come from the activation of the rotameric degrees of freedom. [*8]

7

THEORETICAL MODELS

When searching a complete description of the phase behavior of lipid bilayers, a number of possible

degrees of freedom, interactions, and external conditions should be taken into account. Still, since

only the main transition is in concern, all the factors are not likely to play a crucial role. In this chapter

we first briefly consider the general features of a satisfactory MLBPT model and later on introduce a

simplified one, which still reflects some key characteristics of the MLBPT.

GENERAL REMARKS ON MODELS AND FACTORS INVOLVED IN MLBPT

For the main phase transition that occurs primarily due to the rotameric degrees of freedom, it is

necessary to consider the role of the hydrocarbon chains. While on one hand one could compare

such a system with other “long-chains” systems like alkanes, the case of lipid bilayer differs

significantly due to lack of symmetries. The hydrophobic interaction, for example, assures an average

perpendicular-to-the-bilayer orientation for every lipid molecule with the headgroups positioning

approximately within the bilayer surface. However, in the limit of very long alkyl chain length the

MLBPT indeed resembles the melting transition of alkanes. Nevertheless, it should be mentioned that

the enthalpy changes are roughly twice as large as in the bilayer case due to the anisotropy.

How should the rotameric degrees of freedom be treated? Although, in principle, a continuous rotation

between the C-C bonds would be required, extensive polymer studies have lead to a more simple

treatment by allowing [Fig. 7] only one trans bond and two degenerate gauche states at 120 degrees.

(for the reason that the conformations at intermediate angles are much less populated because they

have energies large compared to kT in contrast to the energies of the trans and gauche states which are

comparable with kT).

Since the conformation of the lipid molecules is strongly dictated by the water content, we expect the

next key factor for the MLBPT is the hydrophobic interaction. However, despite its crucial role in

forming a bilayer, once it is formed, other interactions become more important and the effect of water

stays more or less the same (due to small changes in the area per molecule, that is the case even for

different phases). The only important consequence of a free water presence is the ability of a

membrane to undulate which increases dramatically the vertical and lateral mobility of the lipids.

Consequently, the phases can be very much different if a lipid bilayer is dry, hydrated or exposed to

excess of free water.

Figure 7: A given bond sequence (left) and its projections in the cases of gauche and trans conformations. [*10]

8

Among various interactions that govern the phase phenomena, excluded volume or steric interaction

is not to be forgotten. Its crucial role in the transition can be understood in the following way. At low

temperatures, nearly all chains are in all-trans state, but as the temperature rises, the chains begin to

bump into each other and the steric interaction becomes important. Truly, a collective behavior would

become impossible if the chains were not able to interact with each other and there would not be a

discontinuous-looking change in order but merely a smooth and slow activating of the rotameric

degrees of freedom. The reason is that a change in conformation of a single steric chain forces the

neighbouring chains to react and also change their state. At low temperatures, not enough thermal

energy is available to change the rotameric state of all the neighbours, hence no collective rotameric

response can be triggered. In contrary, at temperatures high enough the rotamers can behave in a

collective fashion. This gives us some qualitative insight into the MLBPT.

From all the other interactions besides the steric one, the van der Waals force is most significant and

exceeds the hydrophobic effect for an order of magnitude [*9]. It is usually presented as being

repulsive at short distances and attractive elsewhere, but since the repulsion may already be included

in the sterical contribution a single power function of distance can be chosen.

There are also many other interesting and measurable effects known, but they play a less major role.

The experiments show, that the transition temperature also depends on the type of headgroups.

Nevertheless, even though the difference is biologically significant, the changes are quite small on the

absolute scale. Similarly, in the most common case of two-tailed lipids the fact that both chains are

tied together in a pair leads to no drastic consequences. Less negligible effect is the presence of a

double bond, which can lower the transition temperature for as much as 60 K, when the unsaturation

occurs in the centres of the hydrocarbon chains [*1]. Another curiosity is the so-called even-odd effect,

characterized for example by the non-monotonous alteration of transition temperatures when

lengthening the hydrocarbon chains by one C-atom [*11].

SOME EXAMPLES OF MODELS

In order to introduce a physical model, a reasonable form of the Hamiltonian should be found first.

The three main contributions [*9,12] to the Hamiltonian of the system can be proposed to be the

internal energy part, the part of dispersive forces between the molecules and the one of the repulsive

steric interaction:

H = Hinternal + Hdispersive + Hsteric. (Eq. 1)

Even though we have already neglected many contributions that are not likely to be tractable, it is in

general still not possible to solve this problem exactly, due to a large number of degrees of freedom.

One possible approach is to undertake the molecular field approximation. This mean-field method

neglects the fluctuations in the value of the local molecular field near the phase transition, but on the

other hand, can be considered and compared with experimental results with no further

oversimplifications quite successfully.

9

In [*11, 15, 16], the author assumes the following forms for the contributions to total energy.

Firstly, the sole contribution to the internal energy of the system is taken to be due to the rotameric

degrees of freedom:

Hinternal = Hrotameric. (Eq. 2)

It would be possible to approximate the rotameric energy of a molecule simple as the number of

gauche bonds times the energy of a gauche bond; however, in the present model a more detailed form

was chosen:

∑=

−+=N

i

iirotameric EEH4

10 ),( ξξ (Eq. 3)

for one chain of length N, with E0 for the energy of the first three chain segments and different

energies for various combinations of consequent bonds: E(ξ,t), E(t,g1,2), E(g1,2,g1,2) and E(g2,1,g1,2).

Here, t stands for a trans bond and gi denotes either of the two gauche bonds.

Secondly, the contribution of dispersive forces between molecules was included in a mean molecular

field Φ as:

Hdisp = – Φ·(ntr/n)∑i

21·(3cos2

θ – 1). (Eq. 4)

With this expression, known from the Maier-Saupe theory [*31], the anisotropic part of the interaction

energy is described to the lowest order. An extra (ntr/n) factor is introduced here to give the correct

order in the ordered phase, but it has no major affects on the results presented later on. The strength of

the molecular field Φ depends on the average order of the system:

Φ = V0 ⋅ ⟨⟨⟨⟨ (ntr/n) ∑ −i

iθ )cos( 212

23 ⟩⟩⟩⟩ (Eq. 5)

the coupling constant V0 is determined from the freezing energy of polyethylene, which is a similar

long chain hydrocarbons system. In the averaged expression we recognize the order parameter S,

defined as an ensemble average of 21 (3cos2

θi – 1), where θi represents the angle between the i-th bond

and the axis of anisotropy which is in our case the normal to the membrane.

Finally, the steric part of the Hamiltonian is included by introducing the externally imposed lateral

pressure P and the area per lipid molecule A:

Hsteric = PA. (Eq. 6)

Writing all together, we have:

H = Hrotameric + Hdispersive + PA. (Eq. 7)

From statistical mechanics, the expression for the partition function Z gives

∑=ons.conformati all

Z exp[–H(Φ,P)/kT] (Eq. 8)

10

and the order parameter S is expressed as:

( ) [ ]

[ ]∫

∫

⋅Φ

⋅Φ⋅−

=1

0

1

0212

23

)(cos/)(exp

)(cos/)(expcos

θ

θθ

dkTS

dkTS

S (Eq. 9)

Analogously, the molecular field obeys the following equation:

Φ = ∑ons.conformati all

{[(ntr/n)∑ −i

θ )cos( 21

12

23

] · exp[–H(Φ,P)/kT]}/Z. (Eq. 10)

These equations are solved numerically, by generating all possible conformations of a single lipid

chain with N molecules. By solving the system for the order parameter S, the dependencies depicted in

[Fig. 8] are obtained.

Besides the good matching between the calculated and measured order parameter, such a model gives

a satisfactory quantitative agreement with experiments also for other quantities not treated in the text

(e.g. for the transition temperature or latent heat).

A model of the MLBPT can be simplified with chain models as proposed by J. F. Nagle [*7, 12–14].

They have a useful property of being equivalent/isomorphic to dimer models that are known from

other physical problems (e.g. physisorption) and can be solved analytically. In the remainder of the

chapter, we first present one of the dimer problems, with no obvious connection to the bilayer problem

but with known solutions. Afterwards, we introduce a chain model that is appropriate to describe

chains in a lipid bilayer. Finally, we establish the correspondence between the dimer model and the

chain model, and obtain the solutions for the latter from the known ones of the former.

Figure 8: Calculated order parameter (averaged over the chain length) as a function of temperature for different values of lateral pressures (in 10-5 N/cm) (left). Right side: order parameter, at 28ºC, shown as a function of a position along the chain for different values of lateral pressure (in 10-5 N/cm). The solid line represents experimental data by Seelig and Niederberger. [*16]

11

If we consider such a dimer model on a brick lattice, depicted in dashed lines in [Fig. 9], a dimer (i.e.,

two-atomic) molecule can be deposited on a brick-like crystal lattice in two ways: either horizontally

(h) or vertically (v). The important fact is that the number of all possible coverings of the lattice with

dimer molecules and consequently the corresponding partition function Z can be expressed

analytically. Moreover, this can also be done if h and v configurations are not energetically equivalent

so that different activities (zh = exp[–εh/kT] and zv = exp[–εv/kT]) could be assigned to them. The

equation determining the partition function Z, then turns out to be [*7]:

∫∫=⋅ππ

φθφθπ

2

0

2

02

)),,,(ln(det8

1),(ln

1vhvh zzMddzzZ

N, (Eq. 11)

where N is the number of dimers, θ and φ are integrational variables, and matrix M can be calculated

for every specific model following a simple recipe [*7,17]. The above formula can be generalized

directly if there are more than two activities zi.

We can now use the analogy between the dimer model on a brick lattice and a chain model on a

triangular lattice in order to create a simple description of the MLBPT [*12-14]. The proposed model

[Fig. 9, filled circles on solid lines] incorporates some basic features: firstly, the steric interaction is

included since at most one chain can pass through every lattice site, and secondly, different possible

directions of bonds are allowed. Nevertheless, it also possesses some undesirable features due to

simplifications. For example, the chains are taken to be infinitely long, in addition to non-real

assumption that lipids are not allowed to leave their vertical columns introducing a vertical anisotropy

in the space. Not to overlook, different order/disorder aspects mentioned in Chapter 2 also cannot be

implemented in the chain model and no volume changes are allowed.

In establishing the correspondence between the chain model of the lipid chains and the aforementioned

dimer model on brick lattice (with the analytically expressed solution for the partition function), we

assume that a gauche bond in the chain model corresponds to a horizontally placed molecule in the

dimer model and a trans bond in the chain model corresponds to a vertically aligned molecule in the

dimer model. We realize that for every state of a chain there is a corresponding analogous dimer state.

Furthermore, the zh and zv activities from the dimer model can be accounted with the activity of a

Figure 9: Chain model on a triangular lattice (dots joined with solid lines and a dimer model on a brick lattice (dashed lines); as explained in text, the models are isomorphous. [*12]

12

gauche bond, x = exp(–εg/RT), and the activity to the activity of trans bonds, y = exp(–εt/RT). Both

models are therefore statistically isomorphic and can be treated equivalently.

From above, it follows that the partition function Z of the chains can be calculated using the

aforementioned formulas for the dimer case.

Since εg and εt are fixed values (εt can be set to 0 leading to y = exp(–εt/RT) ≡ 1, and εg ≈ 2kJ/mol), the

only free parameter is the temperature. By calculating the relative gauche bond density

ρx = (x/N)[d(lnZ)/dx] (Eq. 12)

at different temperatures, we can detect the no-gauche-to-many-gauche state transition, which can be

correlated to order-disorder transition. Namely, for x < 0.5 (i.e., for temperatures T < εg/Rln2) the

calculations give ρx = 0, which describes a phase with no gauche bonds, while for higher temperatures

ρx is a non-zero function of the form:

ρx = π-1 arccos(1/2x); (x > 0.5). (Eq. 13)

The density of gauche bonds is a continuous function of x but its first derivative diverges as x

approaches xC = 0.5 from above. Since the internal energy of rotamers is simply proportional to ρx, the

model predicts a divergence of the specific heat with T from the T > TC side.

Such asymmetrical behavior is typical for many dimer models and appears largely due to the steric

interaction. This phenomenon cannot be dismissed as a pure artifact of the model not relevant for real

systems, for it shows some resemblance with experiments [*18].

Concerning the narrow possibilities of having only one free parameter in the developed model, we

briefly mention that a modification of such a model can be made by introducing vacant sites. An

activity y = exp(–δ/RT) is then assigned to them and the free parameter δ allows the concentration of

vacant sites and consequently the density of a system to be varied. Therefrom, the van der Waals

contribution can also be included as some power function of the density, which yields a more realistic

model than in the case when only the steric interaction is considered.

The presented numerically solved molecular field approach and the exactly solvable dimer model are

well complementary with each other. While the former gives good quantitative agreements with the

experiments, the latter offers more qualitative insight into the key features of the problem. Besides the

two presented models, also other approaches, including Landau theory [e.g., *19] have been adopted.

In addition to more general modeling, models dealing with quite specific factors (such as different

length of the hydrocarbon chains [*20]) were proposed.

13

EXPERIMENTS

CALORIMETRY AND DILATOMETRY

Thermodynamically, calorimetric studies are of primary importance for examination of phase

transitions. In addition, discontinuous volume changes measured by dilatometry can also account for a

phase transition.

From the experimental curve [Fig. 10] obtained by differential scanning calorimetry (DSC), one can

conclude directly whether the transition is discontinuous or not, estimate the temperature interval on

which it takes place and compare the magnitudes of different transitions.

Keeping in mind a simple model, the magnitude of enthalpic changes can be used to estimate the mean

number of gauche bonds and check whether the largest contribution to the MLBPT is due to rotameric

degrees of freedom. Since the measurements [*9] have given a transition enthalpy of around 40 kJ/mol

for synthetic PC lipids and one mole of gauche bonds costs about εg = 2 kJ/mol (for trans bonds, we

take εt = 0), the expected number of gauche bonds would be ng = 40/2 = 20, if the contribution were

solely due to rotameric degrees of freedom.This seems quite large, regarding 28 such degrees in a

DPPC molecule which gives the portion of gauche bonds 20/28 ≈ 0.71. Compare this to the portion of

gauche bonds as calculated from the Boltzmann distribution (for T~300K):

47.0)1/()()2/()2(/ /

21////

states all

/

st. gauche

/≈+=⋅+⋅=

−−−−−−− ∑∑ RTRTkTNkTNkTNkTNkTN ggAgAtAgAiAg eeeeeeeεεεεεεε ,

which is less than a half. Therefore, a more likely explanation is that when the collective disordering

takes place also the expansion of the membrane against the attractive interactions should represent a

considerable contribution to the enthalpy. Indeed, this assumption has been confirmed by dilatometric

measurements [9*].

X-RAY MEASUREMENTS

Besides the intriguing qualitative idea of an order/disorder phase transition small angle X-ray

diffraction experiments allows to obtain also structural quantities like area per lipid, various

thicknesses of the bilayer, spacings between the polar head groups, etc. Since the bilayer thickness

depends on the mean length of the hydrocarbon chains which is further a function of temperature, the

rotameric order may be estimated. On the other hand, the distribution function of the head-to-head

spacings should serve as a measure for positional order.

Figure 10: Sample DSC curve; peaks correspond to first-order transitions whereas baseline shift indicates a second order one. [*21]

14

Figure 11: a caricature of a vesicle (in cross-section) prepared from a mixture of ordinary and spin-labelled lipid molecules; the latter are coloured in magenta.

EPR SPECTROSCOPY

Spin labeling EPR experiments have proved to be very

useful in investigating the phase behavior of lipid bilayers.

Spin labeled molecules with a paramagnetic centre attached

and incorporated into the membrane are used to report about

local ordered in the membrane [Fig. 11]. The measured EPR

spectrum [Fig. 12] is a superposition of the responses from

all the spin labels. If the lipid environment is heterogeneous,

this heterogeneity will reflect itself in the multicomponent

nature of EPR spectra.

Since the shape of a spectrum depends on the anisotropy of

rotational motion, the shape analysis offers some qualitative

insight into the local order parameters. This is possible

because the distribution of resonant magnetic fields depends on the order of the system.

Figure 12: EPR spectra of DMPC vesicles measured at various temperatures. [*22]

15

By taking into account the geometry of the sample

(often spherical vesicles), characteristic properties of

the spin label (penetrating depth, partition

coefficient between different phases, etc.) and the

motion of the spin probes [see comments to Fig. 14],

it is possible to extract accurate values of the order

parameter as well as fractions of molecules in

different phases corresponding to different spectral

components from the measured spectra. By

measuring spectral response at several temperatures,

one can detect the phase transition [Fig. 13]. [*2, 23]

OTHER METHODS

Besides the above mentioned, experiments with other methods such as nuclear magnetic resonance,

infrared spectroscopy [*2], sound velocity measurements, and atomic force microscopy have been

performed [*26]. The latter can be realized on hydrated vesicles only in the tapping mode on

supported membranes or lipid monolayers on a substrate. The thickness of a layer was measured for

both cases and it has been ascertained that the holes in the layer became less evident in the hydrated

case. Because of the substrate effects, the results can be regarded as useful but not a priori

quantitatively relevant. Besides classical experiments, molecular dynamics simulations [e.g., Fig. 6]

also give good insight into the microscopic picture of phase behavior.

Figure 14: cone model.

2

coscos

)(cos2

)(cos)(cos2

sin

sin)(cos

)cos( 002

cos

1313

31

23

cos

1

31

cos

1

223

2

0 0

31

2

0 0

223

312

23 0

0

0

0

0

θθ

θπ

θθπ

ϕθθ

ϕθθθ

θθ

θ

θ

πθ

πθ

+=

−⋅=

−⋅⋅

=

−⋅

=−⟩⟨=

∫

∫

∫ ∫

∫ ∫x

xx

d

d

dd

dd

S

Approximating the rotational motions as fast on EPR time scales at physiological temperatures, the magnetic interactions in the spin Hamiltonian can be averaged through fast random rotational motions of the spin label, consequently leading to an effective Hamiltonian [*25]:

SAISgBHHHefef

BhyperfineZeeman

ef ˆˆˆˆˆˆvvvv

+=+= µ

Order parameter S in these EPR measurements is defined as:

⟩−⟨= 1cos3 221 θS ,

where θ is the angle between a C-C bond and the normal to the membrane. S is used to decribe the anisotropy of these rotational motions in efective Hamiltonian. In the so-called cone model [Fig. 14], the probability density for each bond to wobble within an angle interval [0, θ0] is taken to be constant and values θ > θ0 are forbiden. Within this assumption, the connection between S and the maximal tumbling angle θ0 is the following:

Figure 13: Temperature dependence of the order parameter acquired from simulated EPR spectra [*24].

16

BIOLOGICAL SIGNIFICANCE AND APPLICATIONS

Since biomembranes themselves are lipid membranes of huge biochemical complexity (lipid diversity)

with additional components such as proteins, saccharides, etc., phase transition phenomena can have

many important biological consequence. Different studies of model membranes should therefore lead

to a better understanding of cell-membrane structure. An important fact is that biomembranes exist in

the high temperature phase or at temperature near the transition temepratures of various lipids.

In order to prepare model membranes that resemble biological membranes, different constituents must

be used. For example, manys studies have been

performed to explorethe effects sterols (like

cholesterol) and proteins [Fig. 15]. The fairly rigid

molecules of cholesterol are on one hand known to

induce ordering in disordered phases due to the fact

that they successfully isolate lipid molecules’

hydrocarbon chains one from another themselves

and thus preventing collective disordering effects.

On the other hand, the addition cholesterol reduces

ordering in ordered lipid domains, simply for the

excluded volume effect of its relatively large

headgroups. Protein molecules similarly represent a

boundary condition for the order of lipid molecules

in their vicinity [*27]. The experiments have shown that the latent heat of the main transition

decreases linearly with the protein concentration and falls to zero at the critical concentration. Both

mentioned substances are able to shift the transition temperature, smoothen and widen the transition

and even completely suppress it above the critical point.

Biophysical demonstrations of phases in model membranes (not to forget to mention the coexistence

of cholesterol-enriched and cholesterol-poor domains [Fig. 15]) are in favour of nowadays accepted

view of lipid membranes as heterogeneous structures. Also, for the fact that some membrane

properties (such as ion permeability and lateral diffusion) can differ significantly in various phases,

further research could reveal additional information on the functioning of the cell membrane and the

roles of structurally specialized constituents. It has been shown [*28] that a small change in membrane

composition can produce a dramatic change in protein activity; for example, a change of mole fraction

of cholesterol (xc) in DOPC bilayers from 0.25 to 0.27 results in a 3-fold increase of cholesterol

oxidase initial reaction rate and that cut-off like response could serve as an explanation, why xc is

below 25% in many cell membranes. Despite their role not yet being completely clarified, the

heterogeneous domain structure is understood to be of great importance also in other processes such as

signal processing [*29, 30].

Figure 15: Temperature-composition phase diagram for lipid mixture DMPC/cholesterol. [*2]

17

CONCLUSION

Despite having apparent analogues, phase transitions in lipid membranes remain a unique problem,

mostly due to the complex structure of lipid molecules and also the consecutive anisotropy. These

phenomena have been studied with almost all major experimental methods. Many theoretical models

have proved to be successful in describing them, either by giving good agreements with the

experimental results or revealing the significance of the underlying key features and, furthermore,

leading to theoretical progress. The main factors governing the MLBPT on simpler membranes are

quite well understood, however, the problem in its entirety remains a complex one, assures many

aspects that are not likely to be clarified soon, and certainly, much progress has yet to be made

concerning even more complicated systems resembling the membranes of living cells. Although there

are not very many biological processes in which the role of membrane’s phase behavior is already

thoroughly understood and even some obscurities have been present in many areas for a long time,

there is no doubt that its significance is huge. Even though a notable borderline between complex

biological membranes and simpler model systems is not likely to be overcome soon, the apparent

similarity and compatibility of the both assures the bilayer studies to be promissing and biologically

applicable.

18

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