Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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12 Diﬀusion in Membranes 487

hole (void) nearby to which the lipid can move. The hole may arise from

a vacancy left behind by another lipid molecule, or from collective density

ﬂuctuations in a bilayer.

In the case of a two-dimensional system, the free-volume theory assumes

that the single-particle diﬀusion coeﬃcient can be written as [81]

D =



∞

∗

daD(a) p(a), (12.4)

where D(a) is the diﬀusion coeﬃcient inside a free area a,anda

∗

is the

critical free area deﬁned as the minimum value of a that allows a jump from

one site to another (D(a)=0fora<a

∗

). The p(a) is the probability of

ﬁnding a free area of size a, and is written as

p(a)=

− a

∗

exp



−

Γa

− a

∗



, (12.5)

where Γ is a geometric factor and a

is the average area per molecule. Thus

−a

∗

is the average free area per molecule. For constant D(a), one obtains

D = D(a

∗

)exp



−

Γa

∗

− a

∗



. (12.6)

This expression has later been extended to cases where energy ﬂuctuations

are important [23,78].

Although both experiments [23, 64, 82, 83] and simpliﬁed model systems

[51,84] have provided support for the free-volume theory, one should bear in

mind that the critical free area a

∗

is assumed constant in this description,

while in practice the size and the shape of molecules in membranes ﬂuctuate

in time (thus a

∗

depends on a

). Further, it has been proposed that the free-

volume theory is valid only at high densities over a relatively narrow range

of areas per molecule [66, 84]. Nevertheless, the free-volume approach has

been shown to describe diﬀusion speciﬁcally in the liquid-disordered phase of

single-component lipid bilayers [23].

It is worthwhile pointing out that the microscopic veriﬁcation of the free-

volume theory is still missing. One way to clarify this issue and to provide

more insight into the diﬀusion mechanism is to look at the displacement ∆r

made by a tagged lipid during a time interval ∆t. If the idea of a two-stage

diﬀusion process were true, then the corresponding probability distribution

P (∆r, ∆t) would have a relatively narrow peak around ∆r ≈ R

.Thiskind

of experiment would be straightforward to carry out by MD simulations, and

has been employed in the case of benzene molecules diﬀusing inside a lipid

bilayer [76, 77]. For lipids in a bilayer, however, this idea is probably out of

reach at present due to the high computational cost. Perhaps forthcoming

model studies by MD and coarse-grained approaches (of the kind presented

in [46–49]) will clarify this issue.

488 Ilpo Vattulainen and Ole G. Mouritsen

Besides the free-volume theory, there are several theoretical descriptions

that have been suggested to describe lateral diﬀusion under diﬀerent condi-

tions and in a variety of diﬀerent systems. These descriptions are discussed

in a comprehensive manner, e.g., in [23, 78].

Anomalous diﬀusion

Lateral diﬀusion is often assumed to take place in an unrestricted environ-

ment such that the mean square displacement is linear in time. In membranes

this assumption cannot be taken for granted, however. This idea is supported

by a number of experiments [57,85–87] that have revealed how labeled mole-

cules may be temporally conﬁned to some domain or compartment, or how

the mean square displacement may show anomalous behavior [88–90] as char-

acterized by |r(t)|

∼t

with x =1.

Anomalous diﬀusion may be due to a number of reasons (cf. also Chap. 10).

It has been suggested that proteins embedded in membranes may act as ob-

stacles [91] or binding sites [92] and therefore hinder the diﬀusion of molecules

in a bilayer. In this case the diﬀusion may look anomalous at small times. At

large times, though, one expects normal diﬀusion unless the motion of lipids

is bounded. The diﬀusion may look anomalous also in a situation where the

membrane is locally highly curved or characterized by strong out-of-plane

ﬂuctuations. Then, if the diﬀusion of molecules is interpreted by looking at

their trajectories projected onto the average plane of the membrane, it may

resemble anomalous diﬀusion at short times. Also in this case, however, the

diﬀusion is expected to be normal at long times. Another possible scenario

giving rise to anomalous diﬀusion is the heterogeneity of the bilayer. Perhaps

the simplest example in this respect is a binary mixture of two lipids at a

temperature, where one of the lipids is in the ﬂuid (liquid-disordered) and

the other in the solid-ordered phase. A DMPC-DSPC mixture is an example

of this situation [23]. Then in coexistence of the two phases there are both

liquid-disordered-rich and solid-ordered-rich domains, and the lateral diﬀu-

sion in the ﬂuid phase is very rapid compared to the solid-ordered phase.

Thus, if the relative concentration of lipids in the solid-ordered phase exceeds

some critical value, one expects the solid-ordered phase to percolate through

the system, implying that long-range diﬀusion is no longer possible. In bio-

logical membranes including the cytoskeleton network the situation is even

more subtle, as the membrane-associated network may lead to domains in

which the molecules are trapped for a certain (and usually long) period. The

transient conﬁnement associated with this situation reminds one of anom-

alous diﬀusion at small times, while at (very) long times the molecules are

eventually expected to diﬀuse from one domain to another.

We may conclude that anomalous diﬀusion is relatively common in mem-

branes. However, one should keep in mind that in many cases the diﬀusion

properties depend on the time scale studied. The true tracer diﬀusion coeﬃ-

cient characterizing the lateral motion of individual molecules is found only

12 Diﬀusion in Membranes 489

from the long-time limit through (12.1). Thus, if the technique used looks at

times prior to the long-time regime, it provides information of the dynamics

over a given time scale instead of the actual hydrodynamic limit.

Lateral Tracer Diﬀusion in a Lipid-Cholesterol Mixture

Cholesterol is a major component of the plasma membrane of animal eukary-

otic cells, comprising up to 50 mol % of the total lipid content [93]. A signif-

icant understanding of the role that cholesterol plays in the cell membrane

has been obtained from numerous experimental studies of lipid-cholesterol

model membranes [94]. These studies have demonstrated that cholesterol

has a variety of notable eﬀects on the physical properties of lipid bilayers,

including an increase in the bulk bending modulus of bilayers containing

cholesterol [95], and changes in the orientational ordering of phospholipid

hydrocarbon chains [12]. Cholesterol also has a strong eﬀect on the dynam-

ical properties of lipid bilayers, as is exempliﬁed by numerous experimental

studies [62, 83, 96, 97] which have shown how cholesterol aﬀects the rate of

lateral diﬀusion.

We shall here describe some observations for lateral tracer diﬀusion in

lipid bilayers containing cholesterol in order to illustrate how the diﬀusion

coeﬃcient depends both on the phase state and the degree of conformational

order of long-chain lipid molecules. Oﬀ-lattice Monte Carlo simulations have

been used to study lateral diﬀusion [51] in lipid-cholesterol bilayers using a

two-dimensional minimal, strongly coarse grained model system [98,99]. This

study was motivated by two important ideas. First, the model predicted a

phase diagram in full qualitative agreement with the experimental phase

diagram of a lipid-cholesterol system. Second, since the model retains only

the key properties, which are expected to govern the behavior of interest

thereby omitting details of the molecular description which are not relevant

for phase behavior, one obtains an enormous reduction in the computational

cost of the simulations. This makes it possible to study much larger systems

for longer times, a feature which is necessary in order to be able to investigate

diﬀusion behavior over large time scales.

The phase diagram for the lipid-cholesterol model system [99,100] is shown

in Fig. 12.6. Figure 12.7 shows results for the lateral tracer diﬀusion coeﬃ-

cient D

of a lipid chain as a function of cholesterol concentration X

chol

for

temperatures T/T

=0.99, 1.0, 1.01, 1.02, and 1.03, where T

denotes the

temperature of the main phase transition.

It was found that for T ≥ T

, D

generally decreases as X

chol

is increased

from 0 to 0.45. For these temperatures, there is initially a slow decrease in D

for X

chol

< 0.1, followed by a more rapid decrease up to about X

chol

=0.3,

after which the curves tend to level oﬀ. An interesting observation is that

for T/T

=1.0, D

increases slightly with increasing X

chol

.Atthelowest

temperature of T/T

=0.99, D

increases monotonically with X

chol

. Finally,

for any value of X

chol

, D

always increases with increasing temperature.

490 Ilpo Vattulainen and Ole G. Mouritsen

0.0 0.1 0.2 0.3

chol

0.99

1.0

1.01

T/T

ld+lo

so+lo

CHOLESTEROL

Fig. 12.6. The phase diagram for a model of a lipid-cholesterol mixture as a

function of temperature T and cholesterol concentration X

chol

. T

corresponds to

the temperature of the main phase transition. Shown here are the solid-ordered (so),

the liquid-disordered (ld), and the liquid-ordered (lo) phases. Coexistence phases at

intermediate cholesterol concentrations are also illustrated. The molecular structure

of cholesterol is shown in the inset.

Generally it was observed in the simulation studies that increasing the

cholesterol concentration suppresses D

due to an increased conformational

ordering of lipid chains. However, it seems likely that this eﬀect competes with

an increase in the average free area per lipid, which favours an increase in D

The interplay of these two eﬀects leads to the diﬀusion behavior shown in

Fig. 12.7, which in turn is in excellent qualitative agreement with available ex-

perimental results for lipid-cholesterol mixtures. In these systems the lateral

diﬀusion coeﬃcient decreases by a factor of 2 – 3 for increasing the cholesterol

concentration from 0 mol %, when the system is in the liquid-disordered (ld)

phase for T>T

, to 50 mol %, for which the system is in the liquid-ordered

(lo) phase [51, 83]. Furthermore for T<T

, experimental observations are

in agreement with our results in the sense that increasing the cholesterol

concentration increases D

, with a rather abrupt change close to the phase

boundary between solid-ordered and liquid-ordered phases (so-lo) [69, 101].

In addition, the increase in D

with X

chol

at suﬃciently low T in the lo

phase (T ≈ T

) observed in our simulations was also observed by Almeida et

al. [83] in their FRAP study of lateral diﬀusion of DMPC-cholesterol bilayers.

Overall, the model results are in good qualitative agreement with experiments

and furthermore provide support for the free-volume theory [51].

Given the simplicity of the model, the results for phase behavior and lipid

diﬀusion in the cholesterol mixtures are in remarkably good qualitative agree-

12 Diﬀusion in Membranes 491

0.0 0.1 0.2 0.3 0.4 0.5

chol

T/T

= 0.99

T/T

= 1.00

T/T

= 1.01

T/T

= 1.02

T/T

= 1.03

0.3 0.4 0.5

3.5

4.0

Fig. 12.7. Results for the lateral tracer diﬀusion coeﬃcient of lipids in the model

of a lipid-cholesterol mixture as a function of cholesterol concentration X

chol

.The

curves correspond to diﬀerent temperatures as described by T/T

.Theresultsfor

are given in units of σ

,whereσ is the length scale related to the hard-

core size of a molecule (see [98] for details), and t

is the time in Monte Carlo

units. The inset shows details in the data in order to highlight the non-monotonous

behavior at T

ment with experiment. This demonstrates ﬁrst of all that microscopic details

are in many cases irrelevant for the large-scale properties of complex systems

such as lipid membranes. Further, the good agreement between experiments

and the model system serves as an instructive example of how statistical

physics can contribute to studies of membrane systems. The key idea is to

use the concepts and methods of statistical mechanics to simplify complex

systems. In this manner, one can obtain models, which grasp the essential

features of the system with minimal eﬀort. In the present case, we conclude

that the minimal model described in [51,98–100] is suﬃciently well designed

to describe a wide range of physical behavior of lipid-sterol systems.

12.4 Rotational Diﬀusion of Single Molecules

The rotational motion of lipids and proteins is characterized by the rotational

diﬀusion coeﬃcient D

. To this end, let us use (12.1) and replace r with θ.

The angle θ is chosen to represent the desired degree of freedom such as the

rotation of the vector 

from one of the lipid acyl chains to another (i.e.,

the vector from the center-of-mass of the sn-1 chain to the center-of-mass

of sn-2) around an axis perpendicular to the bilayer plane. Then Einstein

492 Ilpo Vattulainen and Ole G. Mouritsen

equation yields

= lim

t→∞

|θ(t) − θ(0)|

, (12.7)

where θ(t) − θ(0) is the overall change in the angle during a time period t

(thus θ is not bounded by −π and π). This deﬁnition is very suitable for

computational purposes, while in experiments it is less useful. The usual

way to circumvent this problem is to look at the decay of time correlation

functions that are written in terms of the Wigner rotation matrices [3, 39].

This approach is used in experiments such as NMR relaxation, and it is closely

related to the measurement of the lipid conformational order parameter [12],

which is one of the key quantities when the structure of lipid membranes is

being investigated.

As a speciﬁc example, one may consider the correlation function

(t)=P

(u(t) · u(0)), (12.8)

where u(t) is some unit vector at time t,andP

(x)=

−

is the second

Legendre polynomial. The unit vector u(t) can be chosen in various ways,

depending on the motion one wishes to examine. One commonly made choice

is to look at the time dependence of the CH unit vector of some methylene

group in an acyl chain. Then C

(t) essentially measures the rate of rotational

motion of this vector.

Once C

(t) has been measured, one has to ﬁt the results to a decaying

exponential, whose decay rate yields a rotational relaxation rate, which is

proportional to D

. Alternatively, one can estimate rotational diﬀusion coef-

ﬁcients by models such as the one developed by Saﬀman and Delbr¨uck [79],

who treated a lipid molecule (or a protein) as a cylinder. Within this de-

velopment, the lateral and rotational diﬀusion coeﬃcients are coupled by

the membrane viscosity. However, as Moore et al. pointed out [70], these ap-

proaches are marred by the fact that molecules in membranes change their

shape in time and therefore they are not modeled well by a cylinder of some

ﬁxed shape and size. Nevertheless, both approaches [(12.7) and (12.8)] have

been used with success to provide valuable information of rotational diﬀusion

in membranes.

As examples of recent studies on rotational diﬀusion, we discuss a the-

oretical and an experimental approach focused on single-molecule imaging.

Firstly, let us consider the MD work by Moore et al. for lipids in a DMPC

bilayer [70]. They found that the rotational motion of individual compo-

nents in a lipid span three orders of magnitude from about 25 rad

/ns to

0.04 rad

/ns. The fastest component corresponded to the rotation of the sn-1

and sn-2 chains (for a vector from the ﬁrst to the last carbon of each chain),

while the rotation of interchain vectors such as 

were found to be the

slowest modes in this system. Moore et al. also concluded that for most of

the components, the translational and rotational diﬀusion are over diﬀerent

time scales and therefore uncoupled.

12 Diﬀusion in Membranes 493

Secondly, we consider a novel method, called single-molecule anisotropy

imaging, that permits a simultaneous measurement of lateral and rotational

diﬀusion on ﬂuorescence-labeled lipids in a bilayer [59]. In contrast to large

bead labels, a single ﬂuorophore has a well-deﬁned transition dipole mo-

ment with respect to its structure. In the case of the head-group label rho-

damine, its dipole is oriented within the plane of the membrane. Under

proper conditions of excitation lifetime, rotational time scale, and obser-

vation time, the rotational diﬀusion can be imaged directly and the rota-

tional diﬀusion coeﬃcient be derived. Harms et al. [59] found for supported

bilayers of POPC D

=0.07 rad

/ns in the liquid-disordered phase and

=1.2 × 10

−9

rad

/ns in the solid-ordered phase.

12.5 Lateral Collective Diﬀusion of Molecules in

Membranes

Like many complex soft-matter systems, membranes are characterized by col-

lective ﬂuctuations that are essentially driven by thermal excitations. Out-

of-plane ﬂuctuations such as undulations aﬀect the shape and stability of the

membrane as is exempliﬁed by the creation (budding) of liposomes [24] when

a part of the membrane separates from the parent membrane, or when a li-

posome breaks into two. The in-plane ﬂuctuations, in turn, describe density

ﬂuctuations in the plane of the membrane, and are related to area ﬂuctu-

ations that may be one of the mechanisms, which lead to pore formation,

thus allowing passive diﬀusion of particles through membranes. Despite its

importance, it is rather surprising that the collective motions in membranes

are not well understood at the moment [102].

Here we deal with a collective aspect of motion related to density ﬂuctua-

tions in membranes. The density ﬂuctuations are characterized by the motion

of particles as local excess density is spread through collective diﬀusion. It

may sound surprising but in many cases it is the collective diﬀusion rather

than tracer diﬀusion that is measured in actual experiments.

12.5.1 Fick’s Laws

So far we have considered diﬀusion in terms of the Einstein equation that is

related to Brownian motion. However, the way how diﬀusion is described in

many textbooks is rather diﬀerent (cf., e.g., (1.1) to (1.8) of Chap. 1). This is

probably due to the historical background since the so-called Fick’s laws were

likely the ﬁrst attempt to grasp the essential ideas of mass transport. They

were formulated already in 1855 [103] by direct analogy with the equations

of heat conduction derived some years earlier by Fourier [104].

The starting point is a postulate that some of the quantities in a system

are conserved variables. In the case of diﬀusion, the conserved quantity is the

494 Ilpo Vattulainen and Ole G. Mouritsen

particle density ρ(r,t). The second necessary ingredient for a hydrodynamical

description that one is looking for concerns the explicit form for the driving

force in diﬀusion. The general assumption in this respect is that, for systems

close to their equilibrium state, mass transport across a certain unit area is

proportional to the gradient of the particle density normal to the unit area.

These two ideas are described by the continuity equation

∂ρ

∂t

+ ∇·j =0, (12.9)

and the phenomenological Fick’s ﬁrst law for the particle current density

j(r,t)=−D(ρ)∇ρ(r,t), (12.10)

where the hydrodynamic assumption of large length scales and long time

scales is already implicitly included. When these two expressions are com-

bined, one obtains

∂ρ

∂t

= ∇·D(ρ)∇ρ(r,t). (12.11)

The emerging expression is a non-linear partial diﬀerential equation, which

unfortunately is not often analytically soluble. For this reason, one usually

makes an additional assumption that D(ρ) does not depend on the density of

the diﬀusing particles, or that this dependence is weak. Then we get a linear

partial diﬀerential equation

∂ρ

∂t

= D∇

ρ(r,t), (12.12)

which is nothing but the usual diﬀusion equation, which can be often solved

exactly to yield the diﬀusion coeﬃcient D.

Based on Fick’s second law, (12.12), D describes the decay rate of a

(small) density gradient. Evidently this process does not characterize the

same thing as Einstein’s relation (12.1), and therefore one makes a clear

distinction between the two diﬀusion coeﬃcients. The diﬀusion coeﬃcient

corresponding to single-particle motion is the tracer diﬀusion coeﬃcient D

while the collective motion of the whole many-particle system is denoted as

the collective diﬀusion coeﬃcient D

. Depending on the physical situation,

may be equal to the self-diﬀusion coeﬃcient (see, e. g., Chaps. 1 and 10)

and D

is sometimes also called the coeﬃcient of chemical diﬀusion (cf., e. g.,

Chaps. 3 and 18) or transport diﬀusion coeﬃcient (Chaps. 10, 18 and 23).

12.5.2 Decay of Density Fluctuations

To gain some further insight into the physical meaning of D

,letusbegin

with ρ(r,t), which describes the density of the lipid bilayer at position r at

time t. As the density of the system ﬂuctuates in time, and we expect the

12 Diﬀusion in Membranes 495

system to be in equilibrium (or close to it), the relevant quantity here is the

density ﬂuctuation δρ(r,t)=ρ(r,t)−ρ ,whereρ  is the average density in

a membrane. Then we deﬁne the density-ﬂuctuation autocorrelation function

S(r, r



,t)=δρ(r,t) δρ(r



, 0). (12.13)

Now we can write Fick’s second law [see (12.12)] for the ﬂuctuation δρ(r,t)

and assume translational invariance without loss of generality. Consequently

we get Fick’s second law for the density autocorrelation function:

∂S(r,t)

∂t

= D

∇

S(r,t). (12.14)

In the hydrodynamic regime (small wave-vector k and large times), and as-

suming for simplicity the system to be isotropic, the corresponding Fourier

transform decays as

S(k,t)=S(k, 0) exp(−k

). (12.15)

Thus S(k,t) provides an interpretation of D

in terms of the decay rate of

density ﬂuctuations. With a suitable small value of k, the collective diﬀusion

coeﬃcient D

can then be extracted from the long-time tail of (12.15).

As a concrete example, one may think of some region in a membrane

in which the local density is obviously larger than the average density in

equilibrium. Then the density gradient drives the system towards equilibrium.

However, since the excess density at this region cannot spread locally, it must

spread over large length scales (small k) in the course of long times, thus

giving rise to a hydrodynamic collective diﬀusion process.

12.5.3 Relation Between Tracer and Collective Diﬀusion

Under general conditions, there is no simple relationship between the two

diﬀusion coeﬃcients D

and D

. This is noticed by considering the motion

of N (identical) labeled molecules in a membrane. Then the Green-Kubo

equation for tracer diﬀusion is given by



i=1



∞

dt v

(t) · v

(0), (12.16)

and the corresponding Green-Kubo equation for collective diﬀusion is

TρNκ



∞

dt J(t) · J(0), (12.17)

which for simplicity’s sake is here given for an isotropic system. The time

correlation function of the total particle ﬂux J(t)=



i=1

(t) is described

in terms of velocities of the particles, the quantity ρ denotes the density of

496 Ilpo Vattulainen and Ole G. Mouritsen

particles studied in this case, and κ

is the corresponding compressibility.

The corresponding relations for nanoporous host-guest systems are (10.24)

and (10.25) of Chap. 10 (mind the diﬀerence in the notations for tracer (self-)

and collective (transport) diﬀusion in the respective communities).

Combining (12.16) and (12.17) yields

= D

Tρκ





i=j

∞

dt v

(t) · v

(0)



i=1

∞

dt v

(t) · v

(0)



. (12.18)

We see that the cross-correlations v

(t) · v

(0) determine whether a sim-

ple expression between D

and D

can be established. Such an expression

emerges if the velocities (displacements) of diﬀerent particles are not corre-

lated at all.Thenweget

= D

Tρκ

. (12.19)

This situation is expected to be reasonable at very low concentrations, since

then interparticle interactions are not important and (12.19) works well. Fur-

ther, since the thermodynamic factor approaches a constant value in this

limit, lim

ρ→0

Tρκ

= 1 [105], one obtains D

≈ D

at low concentra-

tions.

We can conclude that D

and D

are approximately equal provided that

interaction eﬀects between the labeled particles are weak. In practice this

implies that one should look at the dilute regime at which the concentration

of labeled particles is small. Fortunately this is often the case in experiments

such as FRAP measurements where only a tiny fraction of lipid molecules

are labeled. However, several interesting issues remain unclear. For example,

what is the concentration at which the tracer diﬀusion begins to deviate sig-

niﬁcantly from the collective diﬀusion, and how does the compressibility come

into play in collective diﬀusion. The latter question is particularly important

since the isothermal compressibility

Tρ

N

−N

N

(12.20)

is related to particle number ﬂuctuations in the plane of a bilayer. These

ﬂuctuations, in turn, are expected to show anomalous behavior in the vicinity

of phase transition lines. Thus we can expect that D

may provide one with

information of phase transitions and the locations of boundaries between

diﬀerent phases. Further, the compressibility underlines a marked diﬀerence

between the two diﬀusion coeﬃcients, and it is therefore justiﬁed to expect

very diﬀerent behavior of D

and D

under certain conditions. To address

these questions, we will discuss some model results in Sect. 12.5.5.