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

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12 Diﬀusion in Membranes 497

12.5.4 Methods to Examine Lateral Collective Diﬀusion

Computational Approaches

Having established a model, which can be studied through computer simula-

tions, one can determine D

from density ﬂuctuations by (12.15) with small

k (for a system of size L

,oneobtainsk =2π/L). Then the slope of ln S(k,t)

at large times yields D

. This approach is particularly suitable in the case

of discrete models such as those studied through Monte Carlo simulations,

in which velocities are not well deﬁned. In systems with continuous inter-

action potentials such as those studied in MD simulations, one can use the

Green-Kubo approach in (12.17).

The computational cost of calculating D

is nevertheless large compared

to tracer diﬀusion. This is mainly due to the fact that, contrary to tracer

diﬀusion, all (labeled) molecules in a system are needed to provide one sam-

ple for the evaluation of D

. In other respects, the discussion presented in

Sect. 12.3.2 holds here as well.

Experimental Techniques

The Fourier transform of S(k,t) (in time),

S(k,ω)=

S(k, 0)

+(D

)

, (12.21)

is very useful for practical cases, since it can be measured through (quasi-

elastic) neutron scattering experiments, for example (cf. Chaps 2, 3 and 13).

It yields

S(0, 0)

lim

ω→0

lim

k→0

S(k, ω), (12.22)

when the direction in k-space is ﬁxed.

Another widespread technique used to study lateral diﬀusion is the ﬂu-

orescence recovery after photobleaching (FRAP) [60, 61], which has proved

to be a popular means to assess the structure and dynamics of membranes.

In this technique, a certain fraction of lipids in a membrane is labeled by

a ﬂuorescently active probe. A light source is then focused onto a small re-

gion of the membrane to determine the background level of ﬂuorescence, after

which the ﬂuorescently active molecules within the region of interest are pho-

tobleached by a high-intensity laser. Consequently the intensity coming from

the blackened area drops down, but recovers in time as the ﬂuorescently la-

beled molecules in the surrounding area diﬀuse into the region under study.

By measuring the rate of recovery of ﬂuorescence, one is able to determine

the diﬀusion coeﬃcient [29]. Further, the extent of recovery provides informa-

tion of how many molecules are able to move in a membrane, and therefore

FRAP can provide valuable information of regions where the motion of lipids

or proteins is conﬁned.

498 Ilpo Vattulainen and Ole G. Mouritsen

There are reasons to comment on a few subtle issues regarding this ap-

proach, however. First, the spatial resolution of FRAP is typically about

100 nm, the time resolution about 100 ns, and thus FRAP is not able to de-

tect the short-time dynamics of diﬀusing molecules. Second, the experiments

are done over hundreds or thousands of labeled molecules, and therefore they

do not provide a true description of single-particle motion. Rather, they de-

scribe the rate of how fast a large number of interacting molecules ﬁlls the

region of interest.

It is worthwhile pointing out that the FRAP technique is very similar to

the laser-induced thermal desorption method [106] used in surface science for

studies of collective diﬀusion (cf Chap. 7). Another technique closely related

to the FRAP experiment is the Boltzmann-Matano (BM) method [104] (cf.

Sect. 1.11.1 of Chap. 1), which also is commonly used in studies of collective

diﬀusion of particles migrating on solid surfaces ([106,107], see also Chap. 7).

In the BM method, one considers a step-like density gradient which starts

to spread at t = 0, and the concentration proﬁle in time yields the collective

diﬀusion coeﬃcient. In membranes, the same technique could be applied by

labeling a major fraction of lipids and then photobleaching the lipids beyond

some regime (say, x>0). By following the concentration proﬁle of ﬂuo-

rescently active molecules as a function of x-coordinate in time, one could

extract the collective diﬀusion coeﬃcient of the photobleached molecules. Al-

though this idea seems worthwhile, it has not, to the best of our knowledge,

been carried out yet.

Finally, a number of other techniques are often used in studies of lateral

diﬀusion. NMR T

relaxation [62] measures diﬀusion over short distances,

although the results are interpreted in terms of Fick’s laws. QENS techniques

[63–66] are also microscopic as they extend over short times and distances,

and are comparable to current MD simulations. Larger scale properties of

diﬀusion are measured by NMR-spin echo [62] and ESR [67], among others.

12.5.5 Lateral Collective Diﬀusion in Model Membranes

Collective Diﬀusion in Lipid-Cholesterol Mixtures

We have emphasized in the discussion above that tracer and collective diﬀu-

sion characterize diﬀerent aspects of mass transfer. Let us here illustrate by

an example how this diﬀerence is manifested in practice.

We use the same minimal model as in the study of lateral tracer diﬀusion

of lipid molecules in Sect. 12.3.3. Our objective is to examine a situation where

a certain number of labeled molecules are diﬀusing in a bilayer comprised

mainly of non-labeled lipids. The concentration of labeled molecules is varied

such that the total number of molecules in a bilayer remains constant. In the

present model system, this setup is accomplished by considering cholesterol

molecules as labeled ones, while the lipid molecules form the environment in

which the cholesterol molecules diﬀuse.

12 Diﬀusion in Membranes 499

0.0 0.1 0.2 0.3 0.4 0.5

chol

, T/T

= 0.99

, T/T

= 0.99

, T/T

= 1.02

, T/T

= 1.02

Fig. 12.8. Results for the lateral tracer and collective diﬀusion coeﬃcients of choles-

terol molecules in the model of a lipid-cholesterol mixture as a function of cholesterol

concentration X

chol

.TheresultsforD

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

is the time in Monte Carlo units. Results are given at two temperatures as de-

scribed by T/T

,whereT

is the temperature of the main phase transition. Tracer

diﬀusion results are shown by squares (T/T

=1.02) and diamonds (T/T

=0.99),

while results for collective diﬀusion are illustrated by triangles (T/T

=1.02) and

circles (T/T

=0.99).

In Fig. 12.8 we show data for the tracer and collective diﬀusion coeﬃcients

of cholesterol molecules as a function of cholesterol concentration X

chol

.We

ﬁnd that D

≈ D

at small concentrations, as expected, but at larger concen-

trations of cholesterol molecules the deviation between D

and D

becomes

more and more apparent. In the high-temperature phases at T>T

,the

crossover from D

≈ D

to D

= D

takes place around X

chol

=5%,

after which the two diﬀusion coeﬃcients have little in common. In the low-

temperature phase at T<T

, the deviation between D

and D

becomes

clear around X

chol

= 5 – 7 %, where one crosses the boundary between so

and so-lo phases. Perhaps surprisingly, the quantitative diﬀerence between

and D

is not prominent. Instead, the diﬀerence in both cases is most

pronounced in the qualitative features and is somehow related to the phase

behavior of the system.

The purpose of this example is not to provide quantitative understanding

of how large concentrations of labeled molecules one can safely use in studies

of lateral tracer diﬀusion. However, these results show that D

and D

are

distinctly diﬀerent, and this diﬀerence becomes clear even at relatively small

concentrations. In practice, the crossover concentration is likely to depend

500 Ilpo Vattulainen and Ole G. Mouritsen

on the strength and extent of intermolecular interactions, and is obviously

system dependent.

12.6 Diﬀusive Transport Through Membranes

So far we have been concerned with diﬀusional processes that take place

within and along the plane of the membrane sheet. Since membranes serve

their main purpose as barriers for diﬀusion across the membrane in order

to maintain various gradients needed for biological activity, it is of major

interest to understand what determines these barrier properties and how

they may be modulated. The importance of these questions becomes clear

when considering that the main reason for cell death is membrane leakage,

and that the most eﬃcient way to kill bacteria and microbes is to damage

their membranes by lowering their membrane permeability barrier by poking

holes in the membrane by certain peptides. The most important barriers

for live processes in vertebrates include the blood-brain barrier and the skin,

which both are barriers mediated by membranes. One of the most challenging

questions is to ﬁgure out how new and potent drugs can cross these barriers.

Crossing the lipid-bilayer membrane is a complex process since it involves

traversing a stratiﬁed and highly structured interface of both hydrophilic

and hydrophobic nature [108]. If we ﬁrst consider membrane proteins, these

do not diﬀuse randomly across the membrane without being facilitated by

certain enzymes and transporter molecules. In contrast, the lipid molecules,

being amphiphilic, undergo so-called ﬂip-ﬂop diﬀusion that takes the mole-

cule from one leaﬂet of the bilayer to the other. However, since this process

involves transfer of a hydrophilic head group through the hydrophobic in-

terior, it is exceedingly slow and characterized by time scales in the range

from minutes to days. The ﬂip-ﬂop processes that are needed to maintain the

membrane in its functional asymmetric state are controlled by speciﬁc trans-

membrane proteins (‘ﬂippases’). Spontaneous ﬂip-ﬂop processes can however

be enhanced by various solutes that change the trans-membrane proﬁle.

The passive trans-membrane diﬀusion of various molecular species is re-

ferred to as passive permeability. Lipid bilayer membranes are fairly perme-

able to water, gaseous substances like CO

and O

, and small hydrophobic

molecules like benzene, whereas ions and larger molecular species such as

glucose and amino acids as well as peptides pass only very slowly.

The permeation of molecular species across lipid bilayers [109, 110] de-

pends on both the diﬀusion coeﬃcient and the solubility of the permeant in

the membrane. The permeability therefore intimately reﬂects the inhomoge-

neous nature of the membrane, both transversely and laterally. The detailed

motion of permeants in lipid bilayers is diﬃcult to monitor experimentally,

and new insight into the mechanism of permeation therefore mostly stems

from molecular dynamics calculations [40].

12 Diﬀusion in Membranes 501

It has appeared from such studies, in accordance with interpretation of

experiments [111], that the main barrier to trans-bilayer permeation of small

molecules is the region that contains the carbonyl groups which are in be-

tween the head-group region and the hydrophobic region in Fig. 12.2. In this

region diﬀusion is hampered by the low free volume. Furthermore, the highly

ordered lipid acyl-chain segments underneath the carbonyl groups tend to ori-

ent anisotropic permeants. The rate of permeation is strongly dependent on

the molecular size and polarity of the permeant. Deeper into the hydropho-

bic region, small molecules like benzene take advantage of the available free

volume and diﬀuse by a non-Brownian jump mechanism such that the diﬀu-

sion coeﬃcient turns out to be about the same throughout the hydrophobic

bilayer core. In contrast, larger molecules, e.g. the drug analogue molecule

nifedipine [112], undergo simple Brownian diﬀusion and the diﬀusion coeﬃ-

cient is therefore dependent on the position in the bilayer. The properties of

the head-group region, which is about 8

A wide and has maximum density,

are important for the penetration of ionic species into the bilayer. In this

region the water density gets very small and all water molecules present are

involved in the hydration of the lipid polar head groups. This is the bound-

ary zone where impinging ions will have to change their hydration level and

look for water ﬁngers in order to get into the hydrophobic interior of the

bilayer. The central core of the hydrophobic region is the region of lowest

density because of the disordered chain segments. This hydrophobic region

easily dissolves permeating hydrophobic species and can accommodate large

and bulky (drug) molecules.

Turning now to the eﬀect on permeation of the lateral structure of the

lipid bilayer it is obvious that the considerations for trans-bilayer permeation

presented above have to be modiﬁed in order to account for the possibility

that the actual bilayer proﬁle, which a given permeant sees, can vary laterally

across the bilayer [113]. This is particularly important under those conditions

where the bilayer undergoes large ﬂuctuations in local density or composition

due to the proximity to phase transitions and phase-separation phenomena.

In those cases, small-scale structural heterogeneity and lipid-domain struc-

ture as illustrated in Fig. 12.9 are expected to prevail [114, 115]. The small-

scale structure implies the presence of domain boundaries with less favorable

molecular packing conditions. At these boundaries, which act eﬀectively as

defect lines, more free volume is available which in general will enhance per-

meation under the conditions that produce such heterogeneity. This eﬀect is

illustrated in Fig. 12.10(a) [116], which shows that at the main transition, the

lipid bilayer displays an anomalous permeability behavior in the case of Na

permeation through DPPC bilayers. This permeability anomaly is in fact a

fairly generic phenomenon, which is not only found for small cations but also

for a number of other small and larger molecular compounds.

When several diﬀerent lipid species are present in a bilayer membrane, a

more complex phase behavior arises due to phase coexistence as described

502 Ilpo Vattulainen and Ole G. Mouritsen

Fig. 12.9. Lateral structure of lipid monolayers transferred to a solid support and

subsequently imaged by atomic force microscopy. The lateral structure in these

monolayers are similar to that in bilayers. The diﬀerent grey tones reﬂect height

diﬀerences. (a) Lipid monolayer of DMPC in the ﬂuid phase near the phase transi-

tion. Image size is 5 µm ×5 µm. Maximum height diﬀerence is about 5

A. (b) Lipid

monolayer of a binary mixture of DMPC-DSPC near the coexistence region. Image

size is 250 nm ×250 nm. Maximum height diﬀerence is about 2

by the thermodynamic phase diagram. This has some dramatic consequences

for the permeability of the bilayer. Figure 12.10(b) shows the permeability

of a small negative ion (dithionite, S

2−

) [117] through 1:1 DMPC-DSPC

bilayers. It is observed that the permeability displays two peaks corresponding

to the liquidus and solidus phase boundaries. This implies that the binary

bilayers become leaky at the borders of the coexistence region. Furthermore

it is interesting to note that the level of the permeability in the coexistence

region is considerably higher than in either of the two phases. This suggests

that the mixture is not fully separated and therefore possibly not in thermal

equilibrium.

It is possible to take advantage of these special mixing properties of lipid

bilayers to design bilayer systems, e.g. in the form of liposomes to be used for

drug-delivery purposes, with particularly strong enhancer eﬀects in relation to

permeation. An example is a mixture involving short-chain lipids like DC

[117]. The short-chain lipid molecules enhance ﬂuctuations and hence the

lateral heterogeneity of the liposomal membranes leading to dramatic leakage.

This enhancement of trans-bilayer permeation due to small-scale bilayer

structural heterogeneity is expected to be a generic eﬀect and qualitatively

independent of the permeating species and the lipid material. Of course, the

magnitude of the enhancement and the range of parameter values (tempera-

ture, composition, pH, etc.) over which it occurs depends on details, such as

the size, charge, and hydrophobicity, of the permeant, as well as the materials

12 Diﬀusion in Membranes 503

(a) (b)

Fig. 12.10. (a) Relationship between passive permeability of Na

-ions (•)mea-

sured by radioactive tracer experiments and computer simulation of lipid-bilayer

heterogeneity (—). (b) Thermal scan across the phase diagram of a 1:1 lipid-bilayer

mixture of DMPC-DSPC showing the variation of the trans-bilayer permeability of

dithionite ions (S

2−

) measured by ﬂuorescence quenching techniques.

characteristics of the lipid bilayer (thermodynamic phase, polar head groups,

degree of acyl-chain unsaturation, etc.).

Once having established the relationship between small-scale lipid-bilayer

structure and trans-bilayer diﬀusion, it becomes of interest to explore how one

can use this relationship to modify permeation of certain substances, such as

drugs, by manipulating the microstructure [117]. The eﬀect of cholesterol is

a particularly interesting example [118]. In ﬂuid lipid bilayers, cholesterol in

amounts larger than about ﬁve molar percent eﬀectively acts as an acyl-chain

rigidiﬁer hence lowering the free volume in hydrophobic region of the bilayer.

This leads to a suppression of the permeation and therefore a sealing eﬀect

of bilayer. It is interesting to note that cholesterol (or similar sterols) is uni-

versally present in large amounts (about 30 – 50 %) in the plasma membranes

of all eucaryotes.

12.7 Conclusion

Diﬀusion of lipids and other molecules in membranes is an example of the

dynamic nature of soft ﬂuctuating interfaces. This feature is characteristic for

various soft-matter systems that are governed by interactions on the order of

thermal energy. Consequently, understanding the dynamics of membranes is

highly challenging, since one needs to work on a bridge between truly solid

and ﬂuid phases, and since one has to deal with the time-dependent nature of

diﬀusion as well as the wide range of time and length scales associated with

diﬀusion processes.

In the present chapter, we have tried to illustrate some of the key features

of diﬀusion in membranes. Hopefully we have managed to open the door a

504 Ilpo Vattulainen and Ole G. Mouritsen

little and guide your way to work on the topic. It is fascinating that despite

all work done so far, there are major gaps in our understanding of diﬀusion,

ranging from the microscopic molecular limit of diﬀusion mechanisms up to

the large-scale properties of diﬀusion in lateral heterogeneous environments

of large topological complexity, to name just a few examples. The topic is

genuinely interesting, and we wish to encourage all to reveal its secrets.

Notation

a free area in a bilayer

∗

critical free area in a bilayer

average free area per molecule

(t) time autocorrelation function

d dimensionality

D diﬀusion coeﬃcient

D(a) diﬀusion coeﬃcient inside a free area a

collective diﬀusion coeﬃcient

tracer diﬀusion coeﬃcient

rotational diﬀusion coeﬃcient



diﬀusion coeﬃcient of proteins

h height of the membrane

J(t) total particle ﬂux

j(r,t) particle current density

k wave vector

Boltzmann constant



diﬀusion length



vector from site “1” to site “2”

N number of molecules

p(a) probability of ﬁnding a free area of a

P (∆r, ∆t) probability to ﬁnd a displacement of ∆r over a time interval of

∆t

2nd Legendre polynomial

r(t) position at time t

area per lipid

R radius

S(r, r



,t) density autocorrelation function

t time

T temperature

main phase transition temperature of the lipid bilayer

u(t) unit vector at time t

v(t) velocity at time t

(t) velocity of particle i at time t

x exponent

chol

cholesterol concentration

12 Diﬀusion in Membranes 505

Γ geometric factor

∆r particle displacement

∆t time interval

Θ(t) angle in time t

isothermal compressibility

µ viscosity of the bilayer



viscosity of the surrounding ﬂuid

ρ average density

ρ(r,t) local particle density at r at time t

characteristic time scale

φ(t) velocity autocorrelation function

ω frequency

BM Boltzmann-Matano

DNA deoxyribonucleic acid

DMPC dimyristoyl phosphatidylcholine

DPPC dipalmitoyl phosphatidylcholine

DSPC distearoyl phosphatidylcholine

ESR electron spin resonance

FRAP ﬂuorescence recovery after photobleaching

MD molecular dynamics

NMR nuclear magnetic resonance

PC phosphatidylcholine

PLPC palmitoyl-laureoyl phosphatidylcholine

POPC palmitoyl-oleoyl phosphatidylcholine

QENS quasi-elastic neutron scattering

RNA ribonucleic acid

UQ ubiquinone

References

1. H.Flyvbjerg,T.E.Holy,S.Leibler:Phys.Rev.E54, 5538 (1996)

2. H.C. Berg: Random Walks in Biology, expanded edn (Princeton University

Press, Princeton 1993)

3. U. Essmann, M.L. Berkowitz: Biophys. J. 76, 2081 (1999)

4. G.W. Weiss, R.J. Rubin, Eds.: J. Stat. Phys. 30, 2 (1983)

5. A. Einstein: Investigations on the Theory of the Brownian Movement,edited

with notes by R. F¨urth and translated by A.D. Cowper (Dover 1956)

6. D. Needham: MRS Bulletin 24, 32 (1999)

7. S. Ohta-Iino, M. Pasenkiewicz-Gierula, Y. Takaoka, H. Miyagawa, K. Kita-

mura, A. Kusumi: Biophys. J. 81, 217 (2001)

8. J.A. S¨oderh¨all, A. Laaksonen: J. Phys. Chem. B 105, 9308 (2001)

9. D. Voet, J.G. Voet: Biochemistry, 2nd edn (Wiley, New York 1995)

10. G. Cevc, D. Marsh: Phospholipid Bilayers: Physical Principles and Models

(Wiley, New York 1987)

506 Ilpo Vattulainen and Ole G. Mouritsen

11. R.B. Gennis: Biomembranes: Molecular Structure and Function (Springer,

Berlin Heidelberg New York 1989)

12. M. Bloom, E. Evans, O.G. Mouritsen: Q. Rev. Biophys. 24, 293 (1991)

13. J F. Nagle, S. Tristram-Nagle: Biochim. Biophys. Acta 1469, 159 (2000)

14. Lipid Bilayers: Structure and Interactions. ed by J. Katsaras, T. Gutberlet

(Springer, Berlin Heidelberg New York 2001)

15. Structure and Dynamics of Membranes: From Cells to Vesicles,edbyR.

Lipowsky, E. Sackmann (Elsevier, Amsterdam 1995)

16. Biological Membranes. A Molecular Perspective from Computation and Exper-

iment, ed by K.M. Merz Jr, B. Roux (Birkh¨auser, Boston 1996)

17. Fluctuations and Scaling in Biology, ed by T. Vicsek (Oxford University Press,

Oxford 2001)

18. C. Isenberg: The Science of Soap Films and Soap Bubbles (Dover, New York

1992)

19. D.F. Evans, H. Wennerstr¨om: The Colloidal Domain: Where Physics, Chem-

istry, Biology, and Technology Meet, 2nd edn (Wiley-VCH, New York 1999)

20. S. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin, B.W. Nin-

ham: The Language of Shape. The Role of Curvature in Condensed Matter:

Physics, Chemistry and Biology (Elsevier, Amsterdam 1997)

21. M. Hyv¨onen, K.

O¨orni, P.T. Kovanen, M. Ala-Korpela: Biophys. J. 80, 565

(2001)

22. S.J. Singer, G.L. Nicholson: Science 175, 720 (1972)

23. P.F.F. Almeida, W.L.C. Vaz: Lateral Diﬀusion in Membranes. In: Structure

and Dynamics of Membranes: From Cells to Vesicles, ed by R. Lipowsky, E.

Sackmann (Elsevier, Amsterdam 1995) pp 305–357

24. U. Seifert: Curr. Opin. Colloid Interface Sci. 1, 350 (1996)

25. H.G. D¨obereiner: Curr. Opin. Colloid Interface Sci. 5, 256 (2000)

26. O.G. Mouritsen, K. Jørgensen, J.H. Ipsen, M.M. Sperotto: Domains and Pat-

terns in Biological Membranes. In: Modelling the Dynamics of Biological Sys-

tems. Non-linear Phenomena and Pattern Formation, ed by E. Mosekilde,

O.G. Mouritsen (Springer-Verlag, Berlin Heidelberg New York 1995) pp 77–

100

27. A. Einstein: Ann. Phys. 17, 549 (1905)

28. J.-P. Hansen, I.R. McDonald: Theory of Simple Liquids, 2nd edn (Academic

Press, San Diego 2000)

29. M.J. Saxton: Biophys. J. 81, 2226 (2001)

30. J.-P. Bouchaud, A. Georges: Phys. Rep. 195, 127 (1990)

31. J.W.Haus,K.W.Kehr:Phys.Rep.150, 263 (1987)

32. B.J. Alder, T.E. Wainwright: Phys. Rev. Lett. 18, 988 (1967)

33. B.J. Alder, T.E. Wainwright: Phys. Rev. A 1, 18 (1970)

34. S.C. Ying, I. Vattulainen, J. Merikoski, T. Hjelt, T. Ala-Nissila: Phys. Rev. B

58, 2170 (1998)

35. M. Eichinger, P. Tavan, J. Hutter, P. Parrinello: J. Chem. Phys. 110, 10452

(1999)

36. W. Andreoni, A. Curioni, T. Mordasini: IBM J. Res. & Dev.

45, 397 (2001)

37. M.P. Allen, D.J. Tildesley: Computer Simulation of Liquids (Oxford Univer-

sity Press, Oxford 1993)

38. D. Frenkel, B. Smit: Understanding Molecular Simulation, 2nd edn (Academic

Press, San Diego 2002)