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

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12 Diﬀusion in Membranes 477

The ﬂuid-like nature of membranes is the key to their various dynamic

properties. Out-of-plane ﬂuctuations such as undulations [24] are more like a

rule than exception, since their activation energy is of the order of few k

T ’s.

Fluid-like lipid bilayers have no resistance to shear, unless the cytoskeleton is

attached to the membrane. As the molecules are able to pass one another, the

lateral diﬀusion coeﬃcients of individual lipid molecules are relatively high,

typically of the order of 10

−10

–10

−12

/s. Then, as the lipids and proteins

are complex chain-like molecules, they have various internal degrees of free-

dom, and thus rotational diﬀusion characterizing their rotation rates around

the symmetry axes is also important. The decay of density ﬂuctuations, in

turn, gives rise to the so-called collective diﬀusion (related to area ﬂuctua-

tions of the bilayer), while the diﬀusive transport of lipids across membranes

yields another interesting transport process known as the ﬂip-ﬂop.

The lipid-bilayer component of membranes displays a number of diﬀerent

phases separated by phase transitions. In the present context the most impor-

tant transition is the so-called main transition, which takes the bilayer from a

low-temperature solid phase to a high-temperature liquid (ﬂuid) phase. The

solid phase has a two-dimensional crystalline character and in addition the

lipid acyl chains are conformationally ordered. This phase is therefore usu-

ally referred to as the solid-ordered phase. In the ﬂuid phase, the crystalline

structure is lost and at the same time the acyl chains display a substan-

tial degree of conformational disorder. This phase is consequently termed

the liquid-disordered phase. The diﬀusional characteristics of the membrane

molecules are very diﬀerent in the two phases.

Our aim in the next few sections is to provide the reader with some ﬂavor

of the various diﬀusion processes taking place in membranes. Rather than

presenting an exhaustive description of the topic, we prefer a down-to-earth

approach on an introductory level, combined with instructive examples and

highlights of a few recent studies that we consider interesting.

12.3 Lateral Diﬀusion of Single Molecules

Here we discuss the motion of individual molecules in membranes as the

particles diﬀuse laterally in the bilayer.

12.3.1 Lateral Tracer Diﬀusion Coeﬃcient

Let us consider a membrane comprised of two lipid monolayers (see Fig. 12.2).

Additionally, we may imagine that there are proteins embedded in the mem-

brane, and all these molecules are allowed to move in the plane of the mem-

brane. For simplicity’s sake the membrane is assumed to be ﬂat (zero curva-

478 Ilpo Vattulainen and Ole G. Mouritsen

ture).

Furthermore, we can safely assume that the number of lipid molecules

in both monolayers remains constant, since the time scale of lipids exchang-

ing their positions from one monolayer to another (ﬂip-ﬂop) is of the order

of several seconds or even hours, while the lateral diﬀusion of lipids usually

takes place on much smaller time scales, typically nanoseconds.

A measure of how fast individual molecules move around is given by the

lateral tracer diﬀusion coeﬃcient [27,28] (cf., e. g., also Chap. 10, (10.8) and

Chap. 18, (18.21))

= lim

t→∞

2dt

|r(t) − r(0)|

, (12.1)

where r(t) is the position of the tagged particle (or its center of mass) at time

t,andd = 2 is the dimensionality of a surface.

This deﬁnition is identical

to the original one by Einstein for the motion of Brownian particles in a

solvent [27].

We note that D

is essentially deﬁned as a long-time limit of |r(t)|

,

which is the mean square displacement of a given particle. The tracer diﬀusion

coeﬃcient, however, arises from correlations between consecutive displace-

ments of the particle at small times, as is demonstrated by the Green-Kubo

equation [28] (cf. Chap. 23, (23.15))



∞

dtφ(t), (12.2)

where φ(t) ≡v(t) · v(0) is the velocity correlation function of the tagged

particle (or its center of mass) in terms of its velocity v(t)attimet.

It can be shown that (12.1) and (12.2) are identical and yield the same

diﬀusion coeﬃcient [28]. However, they probe slightly diﬀerent properties of

the dynamics of the particle and therefore complement each other. While

the approach based on the mean square displacement assumes that one has

found the long-time limit (the diﬀusive regime) in which |r(t)|

∼t,the

Green-Kubo equation concentrates on the short-time regime and integrates

over velocity correlations prior to the long-time regime. At long times, beyond

some characteristic time scale τ

, the velocity correlation function φ(t)=0

for t>τ

.Itisobviousthatτ

is closely related to the onset of the diﬀusive

regime in which |r(t)|

∼t (cf, e. g., Chap. 21, Fig. 21.7).

In practice, the velocity correlation function provides a useful quantity

for examining the short-time dynamics through molecular dynamics simu-

lations, thus yielding valuable information of the microscopic details of the

diﬀusion process. Einstein equation, on the other hand, is probably more

useful as a means to determine the actual value of the diﬀusion coeﬃcient

. Besides this, the mean square displacement provides one with a reliable

In the case of liposomes, the calculation for the diﬀusion coeﬃcient has to be de-

termined in a spherical geometry. A further complication arises when the bilayer

is so soft that the diﬀusional motion changes the local curvature.

In the following discussion we assume that r(0) = 0.

12 Diﬀusion in Membranes 479

way to ﬁnd the diﬀusive regime as well as to obtain information of eﬀects

leading to anomalous diﬀusion (cf. Chap. 10). To illustrate these points, let

us argue that |r(t)|

∼t

,wherex is some time-dependent exponent. For

normal diﬀusion, x → 1 at long times, which allows one to ﬁnd the regime

in which the Einstein equation is valid. However, there are also cases where

the heterogeneity of the membrane or large obstacles such as proteins hin-

der or bias the diﬀusion process in such a fashion that x<1evenatvery

long times [29]. Then the diﬀusion process is said to be subdiﬀusive (see

Sect. 12.3.3). The concept of anomalous diﬀusion is generally related to the

so-called Levy processes, which are discussed in more detail in [30, 31].

The diﬀusion of lipids and proteins in a bilayer is essentially a two-

dimensional process, and since we are dealing with a hydrodynamic medium

in which interparticle interactions are mediated by the solvent, it is worth-

while to point out the well-known problem of diﬀusion in two spatial di-

mensions. Under hydrodynamic conditions, φ(t) decays at long times as

φ(t) ∼ t

−d/2

[32, 33], and therefore for a truly two-dimensional system

φ(t) ∼ t

−1

. This seems to imply that (12.2) does not converge and thus

the diﬀusion coeﬃcient is not well deﬁned. However, one should keep in mind

that we are looking at the motion of lipids in a bilayer which is interacting

with the neighboring environment. These interactions imply that the mo-

mentum and the energy of the bilayer are not conserved. Thus, the motion

of lipids in a bilayer is a dissipative process, and the theoretical result above

does not apply in the present case. In brief, as long as anomalous diﬀusion

is not expected, one should ﬁnd |r(t)|

∼t at long times with a constant

slope.

The two approaches mentioned above [(12.1) and (12.2)] are not unique,

although they provide the grounds since they can be considered as deﬁnitions

of the tracer diﬀusion coeﬃcient. Of the other approaches suggested recently,

let us mention the method by which the calculation of D

can be optimized

by a so-called “memory expansion” of particle displacements [34]. Finally,

once D

has been found, one can use it to estimate the diﬀusion length



≡

√

Dτ, which the particle undergoes during a time interval of τ.

12.3.2 Methods to Examine Lateral Tracer Diﬀusion

To start with, we would like to emphasize the special nature of tracer diﬀusion

that should be accounted for in any study of diﬀusion. The tracer diﬀusion

process refers to a case where one follows the motion of some tagged individual

particle.

This is relatively easy in computer simulations of model systems, where

one can keep track of the positions and velocities of all particles one by

one. In experiments the situation is not as simple, however. Many techniques

are based on labeling lipid molecules by attaching rather bulky molecules

such as pyrene to their acyl chains or a colloidal particle to the head group.

The attached molecule has typically some property that allows one to follow

480 Ilpo Vattulainen and Ole G. Mouritsen

its position, ﬂuorescent labeling being probably the most commonly used

technique. In some cases the ﬂuorescence signal from a single lipid molecule or

lipid analogue yields a signal large enough to be detected on the background

of the noise. Most often one labels a certain fraction (such as 1/1000) of all

lipids, however. This leads to two possible problems. First, since the size of the

functional group attached to a parent molecule can be quite substantial, and

its diﬀusion characteristics may be very diﬀerent from those of non-labeled

parent molecules, there is an obvious diﬃculty in interpreting experimental

diﬀusion data. Second, the fact that there are numerous labeled molecules in

a bilayer makes it possible that the trajectories of individual lipid molecules

cannot be followed any further. Consequently, one cannot use the Einstein

equation either. Nevertheless, as discussed later in this chapter, there are

certain ways to avoid this problem.

Computational Approaches

For studies of soft matter systems, the ﬁrst-principles approach including

electronic degrees of freedom feels like a method of choice. However, its com-

putational costs are harsh and imply that it is limited to very small systems

over short time scales, and is therefore not applicable to membrane systems

as such. Current approaches of combining the ﬁrst-principles techniques with

classical molecular dynamics (MD) may provide some relief to this prob-

lem [35, 36].

The classical MD approach ([37, 38], see also Chap. 23), however, has de-

veloped in recent years to a versatile method for studies of lipid and protein

dynamics in lipid bilayers [39–44]. In this approach, all atoms are treated

classically, their interactions are well deﬁned, and the time evolution of par-

ticle positions is determined by solving Newton’s equations of motion. This

approach provides one with a classical but detailed description of the system,

and can yield plenty of relevant information of the structure and dynamics

of membranes at the atomic level. Nevertheless, the price one has to pay is

the computational cost. At present, state-of-the-art MD simulations are for

about 1000 lipid molecules in a bilayer plus a few thousand water molecules,

which allows one to study the system over a time scale of about 100 ns [45].

However, is this enough for studies of lateral lipid diﬀusion? To address this

question, let us ﬁrst approximate the onset of the diﬀusive regime τ

using

aruleofthumbτ

≈ R

,whereR

is the area per lipid in the plane of

the membrane and D

is its tracer diﬀusion coeﬃcient. Using typical values

of R

AandD

=3×10

−11

/s for a single-component lipid bilayer of

DPPC molecules in a liquid-disordered phase [3], we obtain τ

≈ 20 ×10

−9

Thus we have to look at times larger than about 20 ns to make sure that an

individual lipid molecule in a pure bilayer moves over a distance that is equal

to its own size. This time scale should be considered as the lower limit, since

displacements of this size can only barely be regarded as macroscopic ones.

12 Diﬀusion in Membranes 481

In particular in multicomponent systems often characterized by domain for-

mation, the time scale should be far larger since then R

needs to be replaced

by the characteristic size of the domains, which is typically 10 – 1000

A [26].

We can conclude that detailed MD simulations are not the most feasible

approach of studying lateral diﬀusion. Consequently, less detailed approaches

have been suggested. The underlying idea is to simplify the model by replac-

ing an atomistic description with a coarse grained one that retains only some

of the most essential molecular features (see Fig. 12.1). In practice, this means

that the “fast” variables in a system are replaced with stochastic noise and

the interactions are chosen to represent those between coarse grained particles

rather than actual atoms. Recent progress on the ﬁeld is encouraging since

some of the structural [46, 47] as well as dynamic [48, 49] properties of the

membrane have been successfully explored using this idea. However, one can

coarse grain the molecular description even further and describe the system

in terms of a minimal model, in which lipids and proteins are described in a

very simpliﬁed fashion in terms of disks or cylinders, for example. This ap-

proach allows studies of very large time and length scales and may be highly

useful for investigations of processes that are too complicated to be studied

by models including molecular information. Studies have shown [50–52] that

the approach of using minimal models works ﬁne in a number of cases.

Experimental Techniques

The number of techniques which probe lateral tracer diﬀusion directly is lim-

ited. One popular technique is single-particle tracking [53, 54] in which one

attaches a colloidal particle (typically 40 nm in diameter) to the lipid or pro-

tein molecules by proper anti-body functionalization. Then, one follows the

trajectories of individual labeled molecules in bilayers by computer-enhanced

video microscopy, and statistical analysis of a large number of traces allows

one to determine the lateral tracer diﬀusion coeﬃcient in terms of (12.1).

In recent years single-molecule ﬂuorescence labeling assays have been devel-

oped which by ultra-sensitive ﬂuorescence microscopy or ﬂuorescence cor-

relation spectroscopy have permitted traces of single-molecule diﬀusion to

be observed [55–59]. An example of such observations is given in Fig. 12.4

which shows the motion of an individual ﬂuorescence-labeled molecule in a

phospholipid bilayer and the recording of a part of a diﬀusion trace. The

time resolution of such experiments is in the range of 5 ms and the positional

accuracy is around 50 nm [58].

In principle, all other approaches we are aware of are based on follow-

ing some collective property of the whole system, such as the decay rate

of density ﬂuctuations. As we will see in Sect. 12.5, this may lead to stud-

ies of the collective aspects of diﬀusion rather than tracer diﬀusion. Never-

theless, a number of techniques such as ﬂuorescence recovery after photo-

bleaching (FRAP) [60, 61] (cf. also Chap. 16), nuclear magnetic resonance

(NMR) [62] (cf. also Chap. 10), (incoherent) quasi-elastic neutron scattering

482 Ilpo Vattulainen and Ole G. Mouritsen

(a)

(b)

Fig. 12.4. (a) Fluorescent image of a single ﬂuorescence-labeled lipid molecule in a

phospholipid bilayer, and (b) the recording of a part of a diﬀusion trace of a single

lipid molecule. From [58].

(QENS) [63–66] (cf. also Chap. 13), and electron spin resonance (ESR) [67]

have been used to obtain information of single-molecule motion in mem-

branes. The time and length scales probed by these techniques diﬀer sub-

stantially, for which reason they are often characterized as either microscopic

or macroscopic methods (see also Chaps. 1, 9 and, e. g., [68]). We will come

back to this topic in Sect. 12.5, where some of these techniques are described

in more detail.

12.3.3 Lateral Diﬀusion of Lipids and Proteins

Characteristic Diﬀusion Coeﬃcients

One of the compelling ﬁndings in membranes is the great variety of lateral

diﬀusion coeﬃcients, which range from essentially zero to a value of about

−10

/s. To gain some insight into the scales on a practical level, one may

consider a molecule diﬀusing for a period of 10 s at a rate of 10

−11

/s.

Then the diﬀusion length is about 10 µm, which is roughly the diameter of

an animal cell.

12 Diﬀusion in Membranes 483

Diﬀusion in membranes is strongly dependent on phase behavior. As a

practical example one may think of diﬀusion in a single-component bilayer

in the vicinity of the main phase transition temperature T

separating the

solid-ordered phase from the liquid-disordered phase. As the temperature

is changed by a few degrees from T<T

to T>T

, the system goes

through a sudden change from a “frozen and ordered” to a liquid-like bi-

layer, and similarly the diﬀusion coeﬃcient changes abruptly from 10

−20

–

−14

/s [23,69] to about 10

−12

–10

−10

/s [3,23, 69]. The wide range of

diﬀusion coeﬃcients in the liquid-disordered phase has been obtained by a

number of diﬀerent experimental techniques, and even though this may seem

surprising, the diﬀusion coeﬃcient actually depends on the technique used for

the measurement. Furthermore, the diﬀusion coeﬃcient depends on whether

the bilayer is free as in a liposome or supported on a solid substrate [58]

which has in some cases been found to decrease the diﬀusion coeﬃcient by a

factor of 4 – 5.

This observation is essentially due to the diﬀerent time and length scales

probed by diﬀerent techniques. Techniques such as QENS follow the dynam-

ics of molecules in a bilayer over short time scales (less than, say, 1 ns). Thus

they examine the “short-time” behavior often denoted as “rattling in a cage”,

which corresponds to local molecular motion of a lipid within the cage of its

nearest neighbors, over a distance smaller than the size of the molecule. Rigor-

ously speaking, the motion of molecules over such short time and length scales

is not related to the lateral tracer diﬀusion coeﬃcient, since the actual diﬀu-

sion coeﬃcient is deﬁned in the hydrodynamic long-time limit only. However,

one may extract a “time-dependent diﬀusion coeﬃcient” by, e.g., looking at

the time-dependence of the slope of the mean square displacement. MD simu-

lation studies using this approach have led to short-time diﬀusion coeﬃcients

(cf., e.g., (10.9) of Chap. 10) on the order of 10

−11

–10

−10

/s [3,40, 70].

Rather generally, the diﬀusion coeﬃcients get smaller at larger time scales

measured by techniques such as FRAP. This observation is often inter-

preted as evidence of domain formation. Then one expects that |r(t)|

∼t

over a relatively wide time window (say, from 1 ns to 100 ns), yielding an

intermediate-time diﬀusion coeﬃcient, which describes the diﬀusion coeﬃ-

cient within the domain. Only at very long times, however, when the dif-

fusion extends over domain boundaries and protein networks that tend to

slow down the diﬀusion coeﬃcient, one eventually ﬁnds the hydrodynamic

diﬀusion behavior. The time scales that diﬀerentiate these time regimes ob-

viously depend on a system under study, but the point is that macroscopic

techniques (over time scales of microseconds and length scales of micrometers

or more) may yield diﬀusion coeﬃcients that are signiﬁcantly smaller than

the short-time values.

As an example of this scale-dependent diﬀusion behavior, Table 12.1 lists

a few recent results for the lateral diﬀusion of lipids in a DPPC bilayer. This is

one of the most extensively studied single-component lipid systems at present.

484 Ilpo Vattulainen and Ole G. Mouritsen

Table 12.1. Examples of lateral diﬀusion coeﬃcients of lipids in single-component

DPPC (dipalmitoylphosphatidylcholine) bilayers. The time scale over which the

diﬀusion coeﬃcient was determined is also given in some cases. Then “local” refers

to short times while “long range” is over a larger scale.

Technique Time scale T Diﬀusion coeﬃcient Reference

MD < 10 ps 50

C20× 10

−11

/s [71]

MD 0.3 – 0.5 ns 60

C3× 10

−11

/s [3]

QENS 63

C40× 10

−11

/s [72]

QENS “local (max)” 60

C60× 10

−11

/s [64]

QENS “local (min)” 60

C1.5 ×10

−11

/s [64]

QENS “long range” 60

C1× 10

−11

/s [64]

NMR 57

C1× 10

−11

/s [73]

It is clear that the diﬀusion coeﬃcients are reduced at long times, as the ideas

above suggest, although the eﬀects due to domain formation expected at large

scales in many-component lipid bilayers are not included here. Further trends

of lipid diﬀusion have also been discussed in the literature. It has been found

that the diﬀusion coeﬃcient is only weakly dependent on the chain length,

while the number of double bonds (unsaturation) has a pronounced eﬀect on

diﬀusion.

Another remarkable feature associated with membranes is the role of ﬂu-

idity for the diﬀusion of individual molecules. It has been found that in the

liquid-disordered phase the size of the hydrophobic part of the molecule does

not have noticeable eﬀect on the diﬀusion coeﬃcient but it is mostly deter-

mined by the properties of the head group [71, 74, 75]. The recent work by

Pampel et al. is particularly illustrating, as the pulsed ﬁeld gradient NMR

studies of DOPC diﬀusion at 37

C yielded a lateral diﬀusion coeﬃcient of

1.1 × 10

−11

/s. This is in full agreement with the results in Table 12.1

despite the diﬀerences in acyl chain length and saturation between DPPC

and DOPC. This ﬁnding may have major general implications. First of all,

it proposes that labeling does not inﬂuence the diﬀusion coeﬃcients of the

molecules signiﬁcantly, but the labeled molecules move laterally essentially

at the same rate as non-labeled ones. Recent experiments and model calcu-

lations support this view [51, 54, 74]. The same line of thought suggests that

molecules inside lipid bilayers move essentially at the same rate as the lipid

molecules in a bilayer. Again, a number of works are consistent with this

idea. They have shown that molecules close to the headgroups diﬀuse later-

ally approximately at the same rate as the lipids, while molecules near the

center of the membrane diﬀuse 3 – 4 times faster than lipid molecules [8,76,77]

(see discussion below). Further, if the hydrophobic interior of the molecule

and its size are not very important, then one might expect that the lateral

diﬀusion coeﬃcient of integral proteins is roughly as large as the diﬀusion co-

12 Diﬀusion in Membranes 485

Fig. 12.5. Snapshot of an MD simulation [8] of ubiquinone (UQ-10) inside a DPPC

bilayer (water not shown). Here the ubiquinone molecule is “swimming” close to

the center of the bilayer, where the density is substantially lower compared to other

parts in a membrane. Illustration by J. Arvid S¨oderh¨all and Aatto Laaksonen.

eﬃcient of lipid molecules. It turns out that this idea is supported by various

experiments [11].

It is indeed intriguing that the lateral diﬀusion coeﬃcients of proteins

are typically from 10

−14

/s to 10

−11

/s [11, 78]. These are slower than

the diﬀusion coeﬃcients of lipids, but only by a factor of ten or so. Yet

the size and mass of integral proteins are far larger than those of lipids.

Equally remarkable is the ﬁnding that the diﬀusion of proteins is only weakly

dependent on the physical dimensions of the protein. This is demonstrated

in the classical work by Saﬀman and Delbr¨uck [79], who described proteins

as cylinders with radius R, and found that their diﬀusion coeﬃcient can be

described as [79]



4πµh



log

µh



− γ



, (12.3)

where h is the height of the membrane in which the protein is embedded,

µ and µ



are the viscosities of the bilayer and the surrounding ﬂuid, respec-

tively, and γ is the Euler constant. This expression is based on a number of

assumptions (including µ  µ



),whicharediscussedindetailin[78].

Finally, let us brieﬂy discuss the diﬀusion coeﬃcients of molecules inside

lipid bilayers. S¨oderh¨all and Laaksonen studied the diﬀusion of ubiquinone

inside a DPPC bilayer in the liquid-disordered phase through MD simula-

tions [8] (see Fig. 12.5). Ubiquinone acts as a charge carrier inside lipid bi-

layers from one protein to another, and thus the diﬀusion properties are of

486 Ilpo Vattulainen and Ole G. Mouritsen

crucial importance in understanding the charge transfer process. S¨oderh¨all

and Laaksonen found that ubiquinone with a short tail (UQ-et) preferred a

location close to the headgroups of lipid molecules, which is probably due to

the partially hydrophilic nature of UQ-et. The diﬀusion of UQ-et was found

to occur with a rate being essentially the same as the diﬀusion of DPPC

molecules. The diﬀusion coeﬃcient of ubiquinone with a long tail (UQ-10),

in turn, was found to depend on the location of the molecule in a bilayer.

Close to the membrane surface, UQ-10 and lipids diﬀused with similar rates,

while in the center of the membrane the diﬀusion of UQ-10 was found to be

almost four times faster compared to the diﬀusion coeﬃcient of lipids. This

ﬁnding is consistent with earlier studies of benzene diﬀusion inside a DMPC

bilayer [76, 77]. Thus it seems likely that the diﬀusion coeﬃcient of mole-

cules inside lipid bilayers depends on the size and density of voids, which

are greatest in the bilayer center, and that this is somehow related to the

conformational ordering of acyl chains in a membrane.

Above we have discussed the diﬀusion in the liquid-disordered phase. It is

worthwhile pointing out that the picture is likely to be rather diﬀerent in the

solid-ordered phase, where the ordered nature of the membrane comes into

play. In the solid-ordered phase, the diﬀusion of molecules in the center of the

bilayer will be relatively rapid, while the diﬀusion close to the headgroups

will slow down due to the “frozen” nature of the bilayer.

Diﬀusion Mechanisms and Theoretical Frameworks

Despite the signiﬁcant progress in the ﬁeld, the knowledge of the microscopic

mechanisms of how molecules diﬀuse in membranes is scarce. It has been

proposed [65, 80] that the motion of lipids in bilayers consists of two parts:

ﬁrst of local diﬀusion of a lipid molecule at its site, and secondly of jumps

between adjacent sites. This suggestion has been tested by a number of studies

but the overall picture is still cloudy. Essmann and Berkowitz [3] found no

evidence in MD simulations for the two regimes expected in this mechanism,

while Moore et al. found [70] that the two-step mechanism might be valid. The

MD study by Moore et al. dealt with a pure DMPC bilayer, and during a time

scale of 10 ns they found two jump events in which a lipid molecule moved

relatively rapidly over a distance of its size in the plane of the membrane. Most

lipids, however, diﬀused uniformly like molecules in a ﬂuid. This suggests

that the long-range diﬀusion of lipid molecules in pure bilayers (in the liquid-

disordered phase) is a mixture of jumps and uniform motion.

The idea of lateral diﬀusion as a series of jumps from one site to another

is interesting also from a theoretical point of view, since the so-called free-

volume theory [23,78,81] is based on this idea. Originally Cohen and Turnbull

[81] derived the free-volume theory to describe diﬀusion of hard spheres, but

later it has been applied to a number of systems including the diﬀusion of

lipids in a bilayer. In this approach, one assumes that lipids spend long periods

of time at their sites, and long-range diﬀusion takes place only if there is a