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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