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CHAPTER 6

Lateral Diffusion in Membranes

P.F.F. ALMEIDA and W.L.C. VAZ

Universidade do Algarve,Unidade de Ciencias Exactas e Humanas,

P-8000 FARO, Portugal

Present address of P.F.F. Almeida:University of Virginia,

Department of Pharmacology,Charlottesville, VA 22908, USA

1995 Elsevier Science B.V. Handbook of Biological PhysicsAll rights reserved Volume 1, edited by R. Lipowsky and E. Sackmann

305

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

2.1. Diffusion in homogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

2.2. Diffusion in heterogeneous systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

3. Experiments in model systems: comparison with theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

3.1. Diffusion in homogeneous bilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

3.2. Diffusion in two-fluid bilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

3.3. Restricted diffusion of lipids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

3.4. Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

4. Experiments in biological systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

4.1. New techniques: single particle tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

4.2. Restricted diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

306

1. Introduction

The basis of the cell membrane is a bilayer of lipid molecules where proteins areeither embedded (integral proteins) or surface-adsorbed (peripheral proteins) [135].These lipids are elongated, amphiphilic molecules, which, in aqueous media, sponta-neously self-organize so that their hydrophilic headgroups are exposed to water andtheir hydrophobic ‘tails’ are hidden from it. Among the possible structures that resultfrom these requirements is the two-dimensional bilayer membrane, which extendsin a plane perpendicular to the long axis of the lipid molecules. Thermal agitationinduces the movement of lipids and integral proteins in the plane of the lipid bilayer.This thermal movement in two dimensions is identical with lateral diffusion [35].Although a complete treatment must include both rotational and translational diffu-sion, for biological membranes translational diffusion is by far the most important,and we shall restrict our attention to this aspect.

The theoretical and experimental aspects of lateral diffusion of lipids and proteinsin lipid bilayer membranes have been reviewed by Clegg and Vaz [24] up to 1984.The present article is conceived as a sequel of that work. Repetition of discussionswill be generally avoided. For instance, the conceptual questions arising in eachtheory of diffusion in lipid bilayers will be treated in a much more concise man-ner. The reader is referred to [24] for more details. We shall focus instead on thedevelopment in the field from 1984 to 1993.

Two theories of diffusion in homogeneous bilayers were given more emphasis byClegg and Vaz [24]. The first is a continuum hydrodynamic model for diffusion ofparticles the size of which is much larger than that of the solvent. This model isthus best applicable to diffusion of integral membrane proteins in lipid bilayers. Thesecond is a free-volume model, which takes into account the discreteness of the lipidbilayer and is thus best suited to describe lipid diffusion. These two models havebeen used most often and with best results in the interpretation of experimental data.They will be briefly discussed, emphasizing recent work.

A problem not discussed by Clegg and Vaz, except in connection with Stoke’sparadox [71, 117, 174], is that of the definition of a diffusion coefficient in twodimensions. This problem will be reviewed here, not because there was progress inthe matter, but because some confusion has been created about it. Namely, it hasbecome common to speak of microscopic and macroscopic diffusion coefficients,which only apparently have an experimental basis.

It appears to us that the most interesting turn that the field has taken, both theo-retically and experimentally, has been the study of diffusion in heterogeneous mem-branes. These can be lipid–protein membranes with a high protein concentration ormulti-component lipid bilayers where two or more laterally separated phases coexist.

307

308 P.F.F. Almeida and W.L.C. Vaz

When two phases are present in a bilayer, a new, fundamental variable parameterappears, which is the lateral connectivity of each phase. When a phase is self-connected throughout the plane of the bilayer it is said to be percolating. Conversely,if a phase is constituted of disconnected domains it is not percolating. The percolationthreshold of a given phase is the area fraction of that phase above which a percolatingcluster exists. Below the percolation threshold long range diffusion in that phasecannot occur over distances larger than the dimensions of the phase domains. Phasepercolation has been studied in several artificial bilayer systems [168, 169].

In a two-phase system consisting of a percolating fluid phase and a minor, im-permeable phase, tracer diffusion is restricted. The problem, named ‘Diffusion inan Archipelago’ by Saxton [121], is to understand how the presence of obstacles(the impermeable phase) modifies tracer diffusion relative to the all-fluid state of thebilayer. Three cases falling into this general situation have been studied (experimen-tally or theoretically) in membranes: lipid diffusion in solid–fluid lipid systems [6],lipid diffusion in protein–fluid-lipid systems [14], and protein diffusion in concen-trated solutions of proteins in lipid bilayers [129]. In the first case the obstacles aresolid phase domains; in the second, the obstacles are integral membrane proteins; inboth, the tracer is a lipid. In the third case, the tracer and the obstacles are proteins.

A related problem is diffusion in a two-phase system where both phases are fluid.In this case, which has also been studied [7], the tracer molecule can diffuse through-out the entire plane of the membrane no matter which phase percolates. The diffusioncoefficient will, however, depend on the area fractions of each phase present and alsoon which phase is percolating.

This article will focus on some selected aspects, which we feel involve importantgeneral concepts or questions in the field of lateral diffusion in membranes. It isnot intended as an extensive review of all the work on diffusion since 1984. Mostattention is devoted to diffusion in heterogeneous membranes and related problems,a choice obviously biased by our own view of the field.

First, we consider the theoretical aspects that are pertinent to diffusion in lipidbilayer membranes. A large part of this section is based on the review presented inref. [8]. Second, we examine experimental studies in model systems and comparethem with theory. Finally, we consider some studies in biological systems and theconsequences that result for the function of biological membranes in general fromthe structural and dynamical problems associated with diffusion.

As far as experimental methods are concerned, two of them receive most attentionin this article. One method is fluorescence recovery after photobleaching (FRAP)[11, 64], which has proved its usefulness during the last almost 20 years. The otheris single particle tracking (SPT) [46, 48, 131], which although only developed inthe last five years has received much attention and appears very promising. Meth-ods such as FRAP, nuclear magnetic resonance (NMR) [70, 82, 173], or dissipationof electron spin resonance (ESR) signal gradients [31, 134] measure diffusion overlarge distances, whereas methods based on bimolecular reactions, such as Heisenbergspin exchange [97, 115, 153] or fluorescence quenching [43], and neutron scatter-ing [107, 145] measure diffusion over short distances. Single particle tracking canin principle measure diffusion over small and large distances, depending on the time

Lateral diffusion in membranes 309

of observation and spatial resolution of the measuring device, but it is normallyrestricted to relatively small distances.

2. Theory

2.1. Diffusion in homogeneous systems

2.1.1. Definition of a diffusion coefficient in two dimensionsIn a homogeneous, two-dimensional system the diffusion coefficientD can be definedthrough the relation⟨

r2⟩ = 4Dt (1)

where 〈r2〉 is the mean square displacement of a randomly moving tracer, and t istime.

This mean square displacement is the second moment of the Gaussian probabilitydistribution of displacements

C(r, t) =1

4πDtexp

(−

r2

4Dt

)(2)

which is the solution of the diffusion equation [19],

∂C(r, t)

∂t= D∇2C(r, t) (3)

for an instantaneous unit source at the origin at t = 0, where ∇2 is the Laplacianoperator. Hence,

⟨r2⟩ =

∫ ∞0

r2C(r, t) dσ = 4Dt (4)

where dσ = 2πr dr. Whenever the probability density C(r, t) is a Gaussian dis-tribution at long times (large distances) the diffusion coefficient is a well definedquantity.

It has become common to encounter reference to ‘problems’ with the definitionof a diffusion coefficient in two dimensions. Although some difficulties definitelyexist, these arise in well established situations and for reasons which are, for themost part, understood from a theoretical point of view. On the contrary, most of theconfusion in the field has arisen from experimental problems with the methods usedfor determining D, which in some cases measure diffusion over large distances andin others measure diffusion over short distances (see Introduction).

Since some discrepancy has been found between the two sets of methods, it hasbecome frequent to speak of long-range, or macroscopic, and short-range, or mi-croscopic diffusion coefficients [134], and a justification for this fact has often beensought in the aforementioned theoretical problems. It seems to us appropriate to


clarify the situation here because the theoretical and experimental problems arise fordistinct reasons.

We have discussed that part of the problem that may have experimental reasonselsewhere [167]. Briefly, in a discrete medium such as a lipid bilayer, it is notreasonable to expect that the motion of lipid molecules measured over distancesof the order of their sizes [145] could be related to the lipid diffusion coefficient.The detected motion is certainly real and probably corresponds to whole moleculardisplacements rather than to molecular vibrational modes. But these movementsshould be interpreted as the result of local fluctuations and molecular movement inregions of momentarily high free volume. In these regions molecules move muchfaster than on average in the bilayer; but it is the average that corresponds to thediffusion coefficient D, which is a statistical property.

Not surprisingly, if one attempts to calculate a diffusion coefficient from short-range displacement measurements, a much larger value is obtained (up to 2 ordersof magnitude) than that derived from long-range approaches. Using methods thatmeasure movements over distances of only a few lipid diameters (< 10), an apparentdiffusion coefficient may still be calculated, which differs from the long range D

because those distances are probably smaller than the coherence length (ξ) of thedensity fluctuations in a bilayer. Only if the measurement is made over distances� ξ

can the true D be obtained. Einstein [32, 35] used the notion of a sufficiently longtime, which is equivalent. For length scales smaller than the short-range correlationsin a given system diffusion is anomalous.

Turning now to theoretical aspects, three problems have been identified in thedefinition of a diffusion coefficient in a homogeneous system in two dimensions.These are the paradox of Stokes in hydrodynamics, the existence of long-time tailsin the velocity autocorrelation function, and the time dependence of rate constantsfor bimolecular reactions.

Stokes’ Paradox. If we disregard the acceleration convective terms in the Navier–Stokes equation [71, 141], and attempt to solve the hydrodynamic creeping-flowequations for the steady (time-independent), two-dimensional motion, of a contin-uum, viscous, incompressible fluid past a cylinder at rest, we reach the paradoxicalconclusion that it is not possible to satisfy all the boundary conditions (the velocityof the fluid should be constant infinitely far away from the cylinder and zero on thesurface of the cylinder). As Saffman [117] put it for the inverse problem, a steadyforce per unit length on the cylinder induces an infinite velocity. This is the famousStokes’ paradox, which is discussed at length by Lamb [71] (§341–343), Buas [17],Clegg and Vaz [24], and references cited in these works.

The goal of solving the creeping-flow equations is to derive an expression for thefriction coefficient f in the manner of Stokes [71, 141] to be used in the Einsteinrelation for D,

D =kT

f(5)


where k is the Boltzmann constant and T is the temperature, or in the Langevinequation [17, 117]. It should be noticed that this friction coefficient must be aconstant.

There are three approaches to avoid Stokes’ paradox. The first, due to Oseen [71],reintroduces partially the acceleration convective terms and arrives at a solution thatsatisfies approximately the boundary conditions, but still fails to give a constant fric-tion coefficient, which depends now on the velocity. The second approach consists inconsidering the system (fluid) finite; this leads to a diffusion coefficient that dependson the size of the system, a result that appears unsatisfactory, if not unreasonable.The third method, due to Saffmann and Delbruck [116, 117], considers the viscous,two-dimensional medium embedded in another medium, which is three-dimensionaland has a much lower, but finite, viscosity. In this case it is possible to derive anexpression for the friction coefficient (see below).

In conclusion, the difficulties associated with the paradox of Stokes are a con-sequence of the hydrodynamic equations in two dimensions and of the solutionmethods, which avoid dealing with non-linear expressions; however, these difficul-ties do not, in themselves, question the definition of a diffusion coefficient in twodimensions.

Long-time tails. The molecular-dynamic studies of Alder and Wainwright [3, 4] ondiffusion of hard spheres (in three dimensions) and hard disks (in two dimensions)indicated that the the velocity autocorrelation function did not decay in time as anexponential, as generally assumed, but as a power function of time of the form t−d/2

in d dimensions. For d = 2 this long-time tail represents a serious problem. Sincethe diffusion coefficient can be defined as the integral of the velocity autocorrelationfunction, and this integral diverges as t→∞, the diffusion coefficient is not defined!

After the publication of that work, a number of other investigations were reportedon the problem, using computer-simulated experiments [81] or theoretical arguments[37, 110], which supported the initial results. Buas [17] presents a good discussionof the problem and provides more recent references (see also ref. [121]). Two maincriticisms can be made about these investigations. First, the computer simulationsare made in finite systems, of the order of a few hundred particles, and certainlynot in the limit system size → ∞. Moreover, to avoid the interference of thesystem boundaries on the results, the time of the ‘experiment’ has to be chosenrelatively small. Thus, the velocity autocorrelation function may indeed follow amore conventional behaviour as time → ∞. Second, the theoretical treatments arebased on several assumptions that are not necessarily valid [17]. Namely, manyinvestigations are based on linearized hydrodynamic equations (which may presenta problem at the outset, as seen above).

It appears that in Euclidean dimensions > 2 diffusion in fluids of particles inter-acting through hard-core repulsions is normal [54] (at least up to a certain density),although lattice studies indicate the existence of a logarithmic long-time tail in 2 di-mensions [147, 156]. However, some aspects of diffusion in a lattice cannot simplybe applied to diffusion in a continuum, where there are other mechanisms of memorydissipation not present in a lattice. For example, in a lattice it is observed that there


is a correlation between the movement of a tracer and the movement of the vacancywith which it exchanged positions at t = 0 [156]. This question is meaningless incontinuum diffusion because that vacancy can disappear due to small movements ofthe neighboring particles. The lattice structure alters the distribution of areas freeof particles relative to the continuum diffusion case (for example, free areas smallerthan the area of a particle cannot occur in the lattice).

Finally, as indicated by Saxton [121] citing the work of Alder and Wainwright,the long-time tails were observed in hard particle simulated systems and are a con-sequence of local vortices where the energy per molecule is small compared to kT .If the two-dimensional fluid is placed in a three-dimensional heat bath, constituted inthe case of lipid bilayers by the bounding aqueous fluid, the vortices are dissipated bytransfer of momentum between the two fluids. Thus, in experimental homogeneouslipid bilayer systems it is unlikely that apparent abnormal diffusion behavior couldbe attributable to long-time tails arising for the reasons described.

Bimolecular rate constants. The theory of diffusion-controlled reactions in three di-mensions (reviewed by Noyes [99]) indicates that short range correlations can lead tomodifications of the more familiar Smoluchowski equation [139], but a steady-stateis eventually reached. In two dimensions, however, a steady-state cannot be sus-tained [36, 94], and the bimolecular reaction rate ‘constant’ becomes time-dependent(an infinite bulk concentration would be necessary for a steady-state reaction rateto be possible). As noted by Naqvi [96], this situation arises naturally from thesolution of the Laplace equation in two dimensions, which leads to a concentrationprofile depending logarithmically on the distance from the reaction center. It mustbe emphasized that what is time-dependent in this situation is the rate ‘constant’for the diffusion-controlled reaction and not the diffusion coefficient itself. Theexpression proposed by Naqvi for the reaction rate is much more complex thanSmoluchowski’s [139] and has seldom been used in the interpretation of bimolecularreactions, but recent work [87] indicates that it describes experimental data verywell.

2.1.2. Protein diffusion: continuum hydrodynamic theoriesIn these theories [24, 56, 116, 117, 174] the lipid bilayer is represented as a two-dimensional continuum. The finiteness of the lipid molecular structure is ignored.This is not a serious objection, however, if the model is used to describe the diffusionof particles, such as most integral membrane proteins, which are much larger thatthe lipid molecules.

The bilayer is treated as a viscous fluid sheet with a very small thickness h anda viscosity η, bounded on both sides by three-dimensional fluids of much smallerviscosities η1 and η2. The problem becomes three-dimensional (avoiding the Stokes’paradox) by the coupling of the membrane to the bounding fluids. This is achievedthrough transfer of momentum between the fluids as a consequence of the imposedmathematical ‘stick’ boundary condition at the bounding fluid/membrane interfaces.The diffusing molecule is represented as a cylinder of height h and radius R. Its dif-fusion coefficient is given by the Einstein relation [32, 33, 35], D = kT/f , and theproblem is now to calculate f from the hydrodynamic equations.


The linearized Navier–Stokes equation is solved for the membrane sheet and thebounding fluids, assuming the incompressibility of the media. (This assumption,though valid for the three-dimensional bounding fluids, may not be valid for the two-dimensional membrane. In fact, computer simulations have indicated the occurrenceof large ‘holes’ in two-dimensional systems [36]).

The case where η2 = η1 was treated by Saffman and Delbruck [116, 117] and theresult is

f = 4πηh(

lnηh

η1R− γ

)−1

(6)

where Euler’s constant γ = 0.5772. The system can be characterized by the di-mensionless parameter ε = (R/h)[(η1 + η2)/η] [56]. The solution given by Saffmanand Delbruck is valid only for ε 6 0.1. This is appropriate for model membranesbound by water or dilute salt solutions but may not be valid for biological sys-tems where the bounding solutions are often highly viscous. Such is the case ofthe plasma membrane, bound on one side by the cytoplasm, which contains a highprotein concentration, and by the extracellular glycocalix on the other side. Hugheset al. [56, 57] gave, for ε 6 1.0, the solution,

f = 4πηh(

ln2ε− γ +

4ε−ε2

2ln

2ε

)−1

(7)

and, for all values of ε, a numerical solution.Wiegel [174] treated the case of a porous cylinder diffusing within the two-

dimensional viscous sheet. This type of particle could correspond to large, integralprotein complexes in biological membranes. The frictional factor is then

f = 4πηh(

lnηh

η1R− γ +

2

σ2+

I0(σ)

σI1(σ)

)−1

(8)

where I0 and I1 are modified Bessel functions of the first kind and σ = R/√k0, k0

being the thickness by which the membrane fluid penetrates the porous cylinder.Evans and Sackmann [39] considered the effect of weak dynamic coupling of a

membrane to an adjacent solid support. This would apply for example to bilayersor monolayers adsorbed to a solid planar surface. The result obtained is

f = 4πηh( (ε′)2

4+ε′K1(ε′)

K0(ε′)

)−1

(9)

where K0 and K1 are modified Bessel functions of the second kind, and the dimen-sionless parameter ε′ = R(bs/ηh)1/2, bs being a coefficient of friction between themembrane and the support. If a thin layer of fluid of thickness δ � ηh/η1 is insertedbetween the support and the membrane, bs = η1/δ. If δ ' ηh/2η1, then ε′ becomesidentical to ε.


2.1.3. Lipid diffusion: free-volume theoriesThe lipid diffusion coefficient D in homogeneous lipid fluid phases, such as theliquid-crystalline phase (known as Lα or `d) of single-component lipid bilayers orthe liquid-ordered phase (ò found in phosphatidylcholine bilayers containing a highcholesterol concentration), can be derived from free volume theory and its extensionto free-area theory for two-dimensional systems.

On the basis of the free-volume theory of Cohen and Turnbull [25, 50, 154, 155]for diffusion in fluid, glass-forming materials, a free-area model has been developedfor diffusion in two-dimensional systems [7, 24, 43, 89, 161], which was shown toquantitatively describe diffusion in lipid bilayers in the liquid-crystalline phase [161].According to free-volume theory, diffusion of a particle with the size of the moleculesthat constitute the fluid takes place only when a free volume greater than a certaincritical size exists next to the particle. Free volumes smaller than this critical size donot contribute to diffusion, and the problem is reduced to determining the distributionof free volume in the system.

In two-dimensions, the diffusion coefficient is defined as an integral over thedistribution of free-area,

D =

∫ ∞a∗

D(a) p(a) da (10)

where D(a) is the diffusion constant inside a free area a, p(a) is the probability offinding a free area of size a, and a∗ is the critical free area. Areas below a∗ areuseless for diffusion. The probability density p(a) depends exponentially on the freearea,

p(a) = γ/af exp(−γa/af) (11)

where af is the average free-area in the system (af = at − a∗, where at is the totalaverage area per molecule; note that a∗ is a constant, but at and, therefore, af are ingeneral functions of temperature). The parameter γ is a geometric factor that correctsfor overlap of free-areas; it varies between 0.5 and 1. Since D(a) is a constant [25],the expression for the diffusion coefficient becomes

D = D(a∗)

exp(− γa∗/af

). (12)

MacCarthy and Kozak [89], gave a two-dimensional derivation that leads to thesame result if we take a∗ to be identical with the closed-packed area per diskand γ = 1. Almeida et al. [6, 7] have argued that in a lipid system a∗ correspondsto the cross-sectional molecular area in the solid phase and thus considered γ = 1 aswell. Macedo and Litovitz [84] proposed a hybrid equation, which is more generalbecause it includes an energy factor in the diffusion coefficient, and contains theresult of Cohen and Turnbull [25] as a limiting case. That equation, which wasrederived later on the basis of statistical mechanics by Chung [23], has the generalform

D = D′ exp(−γa∗

af−Ea

kT

)(13)


where Ea is the activation energy associated with diffusion. The pre-exponential fac-tor D′ is the ‘unhindered’ diffusion coefficient. If we recall that there are no energyconstraints in the theory of Cohen and Turnbull and compare eqs (12) and (13), wefind that D′ is identical to D(a∗). This pre-exponential factor is related to the gaskinetic velocity, but the exact expressions proposed vary [7, 23, 25, 84, 89, 161].

In the case of a lipid bilayer, the activation energy takes into account the interac-tions of a lipid molecule with its neighbors in the bilayer, the interaction with thebounding fluid, and also the energy required to create a hole next to the diffusingmolecule, whenever this event is locally associated with an energy change [7, 24].

2.2. Diffusion in heterogeneous systems

2.2.1. Definition of a diffusion coefficient in heterogeneous systemsThe definition of a diffusion coefficient in heterogeneous systems may not be possiblein some cases. We then have the so-called anomalous diffusion and a discussion ofthis problem is beyond the scope of this article. We would only like to point out thecases where anomalous diffusion may occur and refer the interested reader to theworks of Bouchaud and Georges [15], Havlin and Ben-Avraham [54], and Staufferand Aharony [144].

For normal diffusion the Einstein relation 〈r2〉 ∼ t (eq. (1)) applies, but for anoma-lous diffusion [54, 144] it is replaced by⟨

r2⟩ ∼ t2/dw (14)

where dw > 2 is the anomalous diffusion exponent. The criterium for normal diffu-sion is still that the second moment of the distribution of displacements be finite forvery large times. In more general terms, diffusion is normal when the central limittheorem applies [15] so that the distribution of displacements approaches a Gaus-sian for a sufficiently long time. More specifically, the distribution of the summedrandom variables (in our case, the displacements in a given time and the waitingtimes between displacements) of the system must not be too broad (finite secondmoment is sufficient), and those variables must not be long-range correlated [15].In a two-phase system a correlation length (ξ) is defined [54, 144] that is essentiallysome length of the typical domains of the minor phase. If the phase domains areuniformly dense (not fractal), for diffusion over distances ` � ξ a tracer does not‘realize’ he is in a two-phase system; diffusion appears normal, with a D corre-sponding to the phase where the tracer was placed. For distances comparable to ξanomalous diffusion is observed. For distances `� ξ diffusion is again normal, withD corresponding to the (long-range) diffusion coefficient in the composite medium.

If the phase domains are fractal structures (readers not familiar with these conceptsmay want to read section 2.2.5 first) the situation is different. Diffusion in percolationclusters at or below the percolation threshold and diffusion in other fractal structuresis anomalous [54, 144]. This is because in these structures there is no characteristiclength scale. If a tracer moves in a finite cluster of length ξ diffusion is anomalous forshort times (`� ξ), whereas for long times ` approaches a constant (ξ), so that D → 0as t → ∞. If the tracer moves in the infinite percolation cluster at the percolation


threshold the correlation length ξ = ∞, and diffusion is anomalous at all lengthscales. (In all these cases the condition `� ξ can never be satisfied.) In the infinitecluster above the percolation threshold ξ is again finite (it basically corresponds tothe length of the typical ‘holes’ in the connected phase). Now, for distances ` � ξdiffusion is anomalous (because the tracer moves in a fractal environment), but for` � ξ diffusion is normal [144], because the details of the phase structure havebeen averaged over those large length scales (the infinite percolation cluster abovethe threshold is not itself a fractal [144]).

In summary, diffusion is normal whenever the movement is measured over dis-tances much larger than the correlation length in the system, if such measurement ispossible.

If some diffusants are trapped within domains and cannot escape even as t→∞while others can move over the entire system, there are two distinct populations ofdiffusants, but diffusion of those that move is still normal (an incomplete recoverywould be observed in a FRAP experiment). On the other hand, anomalous diffusionwill occur if there is a broad distribution of waiting times for particles trapped withindomains [95]. The possibility of occurrence of long-tail kinetics in FRAP studies ofdiffusion in lipid bilayer systems has been recently discussed by Nagle [95].

2.2.2. Diffusion in two-fluid systemsDiffusion studies in a two-phase system where both phases are fluids, or liquids,is probably very relevant to biological membranes. The liquid-disordered–liquid-ordered (`d–ò) coexistence region of phosphatidylcholine/cholesterol binary mixturesrepresents such a system [59, 120]. Recently, significant progress in the determinationof transport properties in two-phase random media has been achieved by researchersin the fields of Physics and Materials Science. Both phases support diffusion (orconduction), but at different rates. Lipid bilayers in the `d–ò two-phase region havebeen used to test experimentally the applicability of these theories to biologicallyimportant systems [7].

A subsidiary aspect of the problem of diffusion in two-phase regions is the possibil-ity of assessing the connectivity properties of the two fluid phases. This componentof the structure of the medium is taken into account by the theories available.

Two reviews by Torquato [51, 151] cover the general theory of diffusion in two-phase systems in detail, which we summarize now, stressing the features most rel-evant to lipid bilayers. For the sake of generality we shall use indexes 1 and 2throughout to designate quantities relating to two phases 1 and 2. The diffusioncoefficient in phase 1 is assumed larger than that in phase 2: D1 > D2. Boundsfor the diffusion coefficient in the two-phase region are obtained by appropriatelyweighing the contributions of the Di in each phase.

Owicki and McConnell [104] derived expressions for diffusion in heterogeneousmembranes of phosphatidylcholine/cholesterol binary mixtures in the solid–fluid co-existence region. This system was modelled as alternating parallel bands of solidand fluid phases (corresponding to the rippled structures experimentally observed inbilayers in this region [26, 68]), where the width of the phase bands is much smallerthan the distance over which diffusion is measured. Diffusion is then anisotropic in


two dimensions. In the direction parallel to the phase bands the diffusion coefficientis

D = f1D1 + f2D2 (15)

where f1 and f2 are the fractions of phases 1 and 2. In the direction perpendicularto the phase bands the diffusion coefficient is

1D

= f11D1

+ f21D2

. (16)

These two expressions give the parallel and perpendicular steady-state electricalconductivities in a medium composed or parallel alternating bands of two materialswith two different conductivitiesD1 and D2. They were first derived by Wiener [175]and represent the upper (eq. (15)) and lower (eq. (16)) bounds on the conductivity ofa two-phase system, which are attained by the alternating band geometry when thecurrent is either parallel (15) or perpendicular (16) to the bands. The two expressionsare obtained even in non-steady state diffusion because of the different ways ofaveraging D1 and D2 to yield the effective diffusion coefficient in the medium [12].However, these bounds are not very useful for a lipid bilayer in the fluid state,because it is extremely unlikely that two fluid phases would adopt such an orderedgeometry.

The `d−ò coexistence region probably corresponds better to a statistically isotropicmixture of phase domains, for which appropriate bounds are also available. Thebounds on the diffusion coefficient in a composite medium correspond to differentconnectivities of the two phases. In general, the upper bound corresponds to ageometry where phase 1 is percolating and the lower bound corresponds to the casewhere phase 2 is percolating.

Hashin [53] showed that the best possible bounds for a two-dimensional, two-phasesystem where the area fractions are the only information available are

Dupper = D11 + f2β21

1− f2β21, (17)

Dlower = D21 + f1β12

1− f1β12. (18)

In these expressions βij = (Di −Dj)/(Di +Dj). In principle, it is possible to assessthe connectivity of the two phases from the dependence of the diffusion coefficientsD in the mixture on the fractions of the two phases present, if the ratio D1/D2is large enough. Experimentally it would be necessary to observe that D shows across-over point from Dupper to Dlower at a particular fraction of one phase.

Milton [91, 92] and Torquato [149] derived improved bounds for statisticallyisotropic media in two-dimensions, and obtained equivalent results. The correspond-ing relations are

Dupper = D1

1 + f2β21 − f1ζ2β221

1− f2β21 − f1ζ2β221

, (19)


Dlower = D2

1 + f1β12 − f2ζ1β212

1 + f1β12 − f2ζ1β212

, (20)

where ζi = fi/3−0.05707f2i [65]. In these equations ζ always refers to the dispersed

phase. In order to obtain the bounds for a system where the the connectivity of thephases is switched, it is only necessary to interchange the indexes everywhere.

Brownian motion simulation studies [65, 66, 152] of diffusion in two-phase sys-tems of arbitrary domain geometry indicate that eq. (19) should be particularly usefulfor the estimation of the diffusion coefficient in two-fluid systems. We shall comeback to this point.

2.2.3. Restricted diffusionWe shall now consider how diffusion of a lipid molecule that is only soluble in afluid phase is modified by the presence integral membrane proteins or solid-phaselipid domains in a bilayer. The lipid molecule under observation (the tracer) cannotmove across proteins and its movement in a solid phase is so slow that for allpractical purposes we can consider that the diffusion coefficient in these regions iszero. Integral proteins and solid domains are therefore obstacles for diffusion ofthe tracer. The situation thus falls under the general problem tracer diffusion in thepresence of obstacles of an impermeable (or insulating) phase, which has receivedmuch attention for a theoretical point of view. In lattice Monte Carlo simulations ofrestricted diffusion, the obstacles are sites that a tracer cannot occupy, whereas in acontinuum problem they can be simulated by impermeable ellipses.

The fractional area (or number of sites in a lattice) where the tracer can move isp; the fractional area occupied by obstacles is c (blocked sites). The tracer performsa random walk in two-dimensions, and the long-time behavior of its mean squaredisplacement 〈r2〉 is related to the diffusion coefficient D [32, 35] by 〈r2〉 = 4Dt(eq. (1)) where t is time. The diffusion coefficient depends on the geometry anddynamics of the obstacles, and the general goal here is to determine this dependence.The absolute value of D is not important for this purpose, but only the ratio D =D/D0 to its value in the absence of obstacles, which in the case of a bilayer is theall-fluid state.

Lattice treatments. Saxton [122, 124] has studied restricted diffusion in a latticeby Monte Carlo simulations. His research has covered the dependence of tracerdiffusion on obstacle concentration, size, and mobility. A tracer performs a randomwalk in a two-dimensional lattice and its mean square displacement is monitored.The diffusion coefficient is obtained from the random walk eq. (1) where t is now thenumber of Monte Carlo steps. In the presence of obstacles, a logarithmic correctionterm is sometimes added to D [122, 156],⟨

r2⟩ = 4Dt+A+B ln t (21)

but this becomes important only at high obstacle densities (anomalous diffusion closeto the percolation threshold).


At constant obstacle area fraction, the diffusion coefficient of a tracer increases asthe obstacle radius increases. In other words, smaller obstacles reduce D more. Thereason is that, since the mechanism of obstruction is collisional, not only the areathe obstacles occupy is important, but also their total perimeter. We shall indicatebelow how this effect arises naturally from an excluded volume argument.

Mobile obstacles are less effective than immobile ones in restricting tracer dif-fusion [52, 122–124]. The ratio γ of tracer jump rate to obstacle jump rate is aconvenient parameter to describe the effect of obstacle mobility. In a triangular lat-tice with point obstacles in the limit γ → 0 the obstacles move so fast that the tracerhas no sense of their positions. At any given instant, a fraction c of sites is occupiedand a fraction p = 1 − c is free; a fraction c of jumps is successful and a fractionp = 1− c of jumps is blocked [122]; thus, D = 1− c. For immobile point obstaclesin a continuum D = 1 − 2c [176], which yields the correct percolation threshold,pc = 0.50. Intermediate obstacle mobilities yield more complex dependencies of Don γ. Saxton [122] showed that, for mobile point obstacles, there is good agreementbetween the Monte Carlo simulations and the theory of correlated diffusion of VanBeijeren and Kutner [156] and Tahir-Kheli and El-Meshad [147]. These authorsdefine a correlation function f (c, γ) related to the diffusion coefficient through

D = (1− c)f (c, γ) (22)

where

f (c, γ) ={[(1− γ)(1− c)f0 + c]2 + 4γ(1− c)f2

0 }1/2 − [(1− γ)(1− c)f0 + c]

2γ(1− c)f0

with f0 = (1 − α)/[1 + α(2γ − 1)]. The constant α depends on the lattice: for atriangular lattice α = 0.2820, for a square lattice α = 1− 2/π [122].

Harrison and Zwanzig [52] considered diffusion in a square lattice where the jumprate from a given site to a nearest-neighbor site fluctuates in time. By numericalsimulation of a master equation in a dynamically disordered lattice, they arrive at thesame conclusions that were reached using the correlation function method [147, 156]concerning the dependence of D on c and γ.

Continuum treatments. In these theories the impermeable (or insulating) phase (areafraction c) is dispersed in a continuum of the other phase (area fraction p), calledthe matrix. In the current literature, there are two basically different approachesto the problem. One is based on computer simulations and arrives at a descriptionof the diffusion coefficient or conductivity which is essentially empirical. This isthe case of the work of Xia and Thorpe [176] where holes (the insulating phase)are randomly punched into a conducting matrix. The second type of approach isan analytical determination of bounds on the diffusion coefficient in such systems,which was discussed above for the case where both phases were conducting. Theregime of restricted diffusion with impermeable obstacles is obtained by making thedispersed phase conductivity → 0 in the final expression.


Again, in the following analysis, the diffusion coefficient D is not an absolutevalue, but always the ratio of D to the diffusion coefficient in the matrix, D0. Letus examine first the simulation type of approach. Xia and Thorpe [176] obtained anexpression for D based on reasonable though not rigorous assumptions. Their resultis interesting because it takes into account the asymmetry of the elliptical inclusions

D =

(1− c

2 + y

2µ+ c2(19 + 4y)

9(2 + y)− (19 + 4y)µ

324µ

)µ(23)

with y = b/a + a/b, where a and b are the major and minor semiaxes of an el-lipse. The dynamical critical exponent µ = 1.3 is probably independent of the staticpercolation exponents to be discussed below [144] (this exponent is often called tin the percolation literature). These results were recently generalized by Garbocziet al. [44] who derived a universal conductivity curve where the variable is not c butx, x = ρL2

eff, where ρ is the number density of inclusions and Leff = 2(a + b) [seesection 2.2.5 (Percolation) for more details]. The conclusion is

D =

[ (1− x/〈xc〉)(1 + x/〈xc〉 − x2/24.97)

1 + x/3.31

]µ(24)

where 〈xc〉 = 5.9 is the average of xc (the value of x at the percolation threshold) overdifferent orientations. For example, for randomly oriented ellipses (with any aspectratio) xc = 5.6. Equation (24) is in good agreement with random walk simulationsand experimental conductivity measurements [44].

Consider now the analytical approach. The theories in question correspond to thelimit D2 → 0 in the equations for the diffusion coefficient discussed above (17)–(20);that is, the phase with lower diffusivity becomes completely impermeable. One ofthe first treatments was given by Maxwell in 1873 [88]. In two dimensions [149],his result corresponds to

D =1− c

1 + c. (25)

Many modern results are improvements upon this equation, but reduce to it in theappropriate limits. An obvious flaw of Maxwell’s equation is that it does not predicta percolation threshold. Improvement on this aspect is achieved by the Bruggeman–Landauer equations [16, 72, 122],

D = (1/2− c)(1− ε) +√

(1/2− c)2(1− ε)2 + ε (26)

where ε is the ratio of the diffusion coefficient in the inclusions to that in the matrix.In the limit ε = 0 eq. (26) predicts pc = 0.50.

As described above, some authors [51, 151] have derived exact bounds for D in atwo-phase system instead of trying to find an expression for the value of D, whichdepends on the particular structure of the phase clusters, information that is most


often not available. When one of the phases is impermeable, the lower bound isalways zero, corresponding to the case where this phase is percolating. Maxwell’sequation (25) is actually one of the simplest expressions for the upper bound. Forimpermeable obstacles, eq. (19) reduces to

D =1− c− pζ

1 + c− pζ(27)

where ζ = c/3 − 0.05707c2 [65]. If we put ζ = 0, this equation yields Maxwell’sresult.

All theories outlined in this section – both simulation-based and analytical – sufferfrom one common weakness: the size of the obstacles does not enter the formula-tion. The reason is that the ‘tracer’ they consider has no size and, therefore, nocharacteristic dimension enters the problem. In the lattice Monte Carlo simulations,this is not the case [124]. Tracer diffusion is sensitive to obstacle radius, except inthe limit of very large obstacles. This arises because of the effective finiteness ofthe size of a ‘point’ tracer in a lattice. The characteristic unit length of the problemis then the lattice spacing. The results of the continuum approach are thus to becompared with the lattice calculations in the limit of of very large obstacles.

In order to apply these results to lipid bilayers where the obstacles are not neces-sarily in the limit of infinite radius, a correction for finite size effects is necessary.Very recently, Kim and Torquato [67] indicated a correction procedure to apply thecontinuum results to systems where the tracer is not much small than the obstacles,for the three-dimensional case. The procedure is based on taking into account theexcluded volume effect arising from the finite size of the tracer. In two-dimensions,this argument requires that we replace c by c (1 + r0/R)2, where r0 is the radius ofthe tracer and R is the radius of the obstacles.

2.2.4. Protein diffusion in crowded systemsThe effect of a high protein concentration in the bilayer on protein diffusion hasbeen reviewed recently by Scalettar and Abney [129]. The self-diffusion coefficientof interacting integral proteins in concentrated solutions in bilayers is altered relativeto its value in the limit of infinite dilution. Three kinds of interactions have beenconsidered [2, 93]: a hard-core repulsion (excluded volume) interaction, a repulsiveplus attractive (‘Lennard–Jones type’) 6–4 potential, and an exclusively repulsivesoft potential. The theoretical hard-core interaction model of Abney et al. [2] agreeswell with the dilute limit regime of the Monte Carlo simulation data of Saxton [122]and Pink [109], which agree with each other over almost the entire concentrationrange of diffusants. As protein crowding increases, the relative diffusion coefficientD follows a smooth, non-linear, monotonically decreasing function of the proteinarea fraction, c, from D = 1.0 at c = 0 to D ' 0 at c ' 0.8 (see fig. 5 of [122]and fig. 3 of [129]). Minton [93] has proposed a different version of this type ofmodel based on free volume and scaled particle theory. His results agree well withthe lattice models [109, 122] for a certain choice of a free parameter and a proteinarea fraction up to 0.3.


Abney et al. [2] also considered the diffusion of integral proteins subject to a6–4 potential composed of a soft, short-range repulsive interaction and a long-rangeattractive interaction. The effect of attractive forces is shown to be extremely small,since the results obtained with the combined attractive and repulsive forces are al-most identical with those obtained by considering repulsive interactions alone [2]. Ingeneral terms, both attractive and repulsive interactions decrease the diffusion coef-ficient, which shows again a monotonical decrease as the protein density increases.

Unfortunately, these theoretical calculations give the diffusion coefficient as afunction of a reduced density of proteins ρ∗ = ρσ2, where ρ is the actual proteindensity and σ is the zero of the 6–4 potential. As indicated by those authors [2], theparameter ρ∗ is not simply related to the protein area fraction c, which is the morefrequently used variable. However, we may estimate the effect of these potentialsby making σ ' diameter of a protein, which leads to c ' 0.78ρ∗. Within thisapproximation, we can conclude that the effect of a soft repulsive (or repulsive–attractive) potential on the diffusion coefficient of proteins in concentrated solutionsis very similar to that obtained with a hard-core repulsive interaction alone (comparefigs 3 and 5 of [2] taking into account the scale conversion just indicated).

Experimental data on this problem is extremely scarce. As pointed out by Abneyet al. [2], the diffusion of proteins as a function of protein concentration has beenstudied in two cases only: that of gramicidin (which is hardly a protein!) [146] andthat of bacteriorhodopsin [106, 160]. These studies were performed about 10 yearsago and have already been discussed by other authors, so we comment on them nowand briefly, and not in section 3.

Comparison with experiment has indicated that models of diffusion including ahard-core repulsive interaction underestimate severely the effect of crowding onprotein diffusion [122]. The same criticism should apply to models using attractive–repulsive potentials [2], in view of the argument presented above. As frequentlysuggested [2, 122] the reason for this discrepancy may be an ordering effect of theproteins on a boundary layer of adjacent lipids, which would result in an increasedeffective viscosity of the lipid bilayer. We have recently developed a model incor-porating this feature [6] and applied it to the description of lipid diffusion in thepresence of obstacles, be they proteins or solid-phase lipid domains, with very goodresults. A similar approach seems appropriate here.

2.2.5. PercolationSince the experimental systems to be discussed are two-dimensional, this sectionwill be restricted to a brief review of percolation theory in two dimensions. Thereare two types of percolation problems: bond and site percolation [144]. We shallconsider only site percolation. In this case a cluster is a group of occupied adjacentsites that are not connected to any other group.

In the bond percolation problem bonds are randomly placed in-between the latticesites. A cluster is then formed by sites that are connected through bonds. We mentiononly two recent papers by Saxton [125, 126] that consider the spectrin network ofthe cytoskeleton in erythrocytes from the point of view of bond percolation. Thatauthor discusses the effect of the spectrin network as a barrier for long-range lateral


diffusion of membrane proteins, as well as its effect on the mechanical resistance ofcells and possible biological consequences of cytoskeleton (bond) percolation [126].

In percolation theory, there are two main branches of research. The first treatspercolation as a second-order phase transition. Major concerns are the behaviorof various average properties, such as the number of clusters and their size, suffi-ciently close to the percolation threshold pc. In particular, the theory attempts todetermine the critical exponents in the functions describing the behavior of theseproperties close to pc. An excellent review of the problem is the book by Staufferand Aharony [144]. The second branch is concerned with the value of pc itself in atwo-phase continuum as a function of the shape of the objects that constitute the dis-persed phase. Particularly useful references are the papers by Xia and Thorpe [176]and by Garboczi et al. [44]. These two perspectives of percolation are discussed inthe next two paragraphs.

Percolation as a critical phenomenon. Consider an infinite two-dimensional lattice.The sites of the lattice are randomly occupied with probability p and empty withprobability 1− p. While the lattice is mostly empty, the occupied sites form clusterswith sizes that increase with p. When the fraction p of occupied sites becomeslarger than the critical value pc (the percolation threshold) an infinite cluster appearsin the plane. For example, in a triangular lattice pc = 0.50, in a square latticepc = 0.59 [144]. The structure of the system can be described by the various averageproperties. Let ns(p) be the number of clusters per lattice site with s occupied sites.The strength of the infinite cluster, which is the probability P that a lattice sitebelongs to the infinite cluster, is given by [144]

P =∑

s[ns(pc)− ns(p)

]. (28)

Notice that P is different from the percolation probability of the lattice. The latter,which will be designated by R(p), is the probability that an infinite cluster exists ata given fractional occupancy p. In an infinite lattice, R(p) = 1 for p > pc, whereasP = 1 only when p = 1. The average cluster size or area S is

S =

∑s2ns∑sns

. (29)

Close to the percolation threshold pc, the cluster numbers ns, the strength of theinfinite cluster P , and the cluster size S depend on a few critical exponents [144].

The strength of the infinite cluster scales according to the exponent β = 5/36,

P ∼ |p− pc|β (30)

and the cluster size S according to

S ∼ |p− pc|−γ (31)


where the critical exponent γ = 43/18. The probability that two occupied sites ata distance r from each other belong to the same cluster is given by the correlationfunction

g(r) = exp(−r/ξ) (32)

where the correlation length ξ (some average distance between two sites of the samecluster) scales according to another critical exponent, ν = 4/3,

ξ ∼ |p− pc|−ν . (33)

In order to apply the results to real systems, it is important to know the behavior ofaverage properties when the lattice has a finite size. Consider a lattice that occupiesa square portion of the plane. The size of the lattice is defined as the length Lof the side of the square; this length is measured in units of the lattice constant,or lattice spacing, λ. The points of the lattice are again randomly occupied withprobability p and we look for a percolating cluster, which spans the lattice fromtop to bottom (we could equally well look at percolation between the left and rightsides). A percolating cluster will still be referred to as the infinite cluster. Thissimply means that the length of the percolating cluster is of the same order as thelength of the system.

Whereas in an infinite lattice the probability of percolation R(p) = 0 below pc andR(p) = 1 above pc, in a finite system this probability is a continuous function whichhas intermediate values in the neighborhood of pc. As the lattice size increases R(p)becomes steeper around the threshold. Thus the slope of R(p) at p = pc appears tobe proportional to L. The exact relation is [144]

R(p) = F[(p− pc)L1/ν] (34)

so the slope,

dR

dp∝ L1/ν . (35)

The exact form of the function R(p) is not known. In general, it is an S-shaped func-tion with an inflection around the percolation threshold, which obeys the boundaryconditions

limp→0

R(p) = 0, (36)

limp→1

R(p) = 1. (37)

Several functional forms have been used for R(p). The integral β distributionfunction was shown to give accurate fits of computer data even for small systemsizes [113] and is frequently used [98, 127], whereas an error function (dR/dpis then a Gaussian) is a good assumption for large system sizes [144, 178]. Other


approximations forR(p) include exponential forms such as a Weibull distribution [55]and analogies to thermodynamics [9]. The use of approximations is acceptable if thenoise in the data is larger than the error involved.

If 〈pc〉 is the average percolation threshold, the width of the percolation transition(or the uncertainty in pc) decreases as the lattice size increases according to theexponent ν∣∣〈pc〉 − pc

∣∣ = ∆pc = kL−1/ν (38)

where k is a constant [144]. Thus 〈pc〉 = pc± kL−1/ν , and the percolation thresholdcan be determined from a plot of 〈pc〉 versus L−1/ν for several values of L, yieldingpc as the intercept at the origin [98, 113, 127, 144, 178].

We have only considered lattices with a random distribution of the occupied sites.In lipid bilayers the interaction energy between molecules can lead to appreciabledistortions in some of the quantities described. The structure and size of the finiteclusters are particularly sensitive to this effect. A complete description of the finiteclusters must include not only the average area, but also the structure and shape.The structure of a cluster may be described by its dimensionality. The area S of acircle or a square varies as the square of its linear dimension φ (the diameter or thelength of the side),

S ∼ φ2. (39)

The dimensionality of these objects is 2, and coincides with the Euclidean dimensionof the space in which they are embedded, the plane. However, this relation does notapply to an object with an arbitrary structure. A familiar case for which it does notapply is that of a snow crystal. The more general relation between the length andthe area of an object embedded in a two-dimensional space is

S ∼ φdf (40)

where df is the dimensionality of the object, which may be fractal [40]. In a randomlyoccupied lattice, percolation theory can predict the fractal dimension of the finiteclusters. Below the threshold (p < pc) df = 1.56, at p = pc df = 1.896, and abovethe threshold (p > pc) df = 2 [144]. Deviations from these values in experimentalsystems are very likely to arise because of interaction energies between molecules.

Continuum percolation. Two aspects of continuum percolation theory, which areparticularly relevant for lipid bilayers, will be considered. The first is the problem ofthe determination of the value of pc as a function of the shape of perfectly insulatinginclusions randomly distributed on a matrix of a conducting phase. The second isthe steepness of the approach to the threshold as a function of the size of the systemor, more precisely, how R(p) depends on L.

Xia and Thorpe [176] have studied the percolation properties of random freelyoverlapping ellipses in a two-dimensional continuum as a function of their aspect


ratio. The aspect ratio of an ellipse is b/a where a and b are the major and minorsemiaxes. They considered a square portion of the plane and gradually increased thenumber of identical ellipses (phase c), which were allowed to overlap, in this region,until the matrix (phase p) ceased to percolate. This point is the percolation thresholdpc (in this case it is the area of the matrix and not that of the ellipses). At any p,the area of the matrix decreases exponentially with the number density ρ of ellipses

p = exp(−Sρ) (41)

where S = πab is the area of an ellipse. Note that c = 1 − p. The percolationthreshold was found to decrease as the aspect ratio increases. This means thatelongated ellipses disconnect the matrix at a smaller area fraction c than circles do.For overlapping circles pc = 0.33, and pc → 1 in the needle limit, b/a→ 0. Theseresults can be summarized [44] by the formula,

pc =

(1

3

)4/(2+y)

(42)

where y = b/a+ a/b. For circles, this gives simply y = 2 and pc = 1/3.Garboczi et al. [44] also defined a new quantity x,

x = ρL2eff (43)

where Leff = 2(a+ b). At pc, xc = 5.6 for all aspect ratios. The quantity x is usefulin the calculation of the normalized conductivity, which was discussed above.

The influence of system size L on the probability of percolation R(p) in a contin-uum has also been studied [78]. Consider a square portion of the plane made of amatrix of an insulating phase and keep increasing the number of partially penetrabledisks (of diameter φ) of a conducting phase. The area fraction of this phase is p, andpc is the fraction of disks at the threshold. The length of the system L is measuredin units of φ and was varied from 5 to 40 [78]. The dependence of R(p) on L isdescribed by a function of the form of eq. (34). The exact value of pc dependson the degree of allowed disk overlap, but it is generally close to 1/3 as given byeq. (42) [78].

3. Experiments in model systems: comparison with theory

3.1. Diffusion in homogeneous bilayers

The translational diffusion coefficient D of lipids or proteins incorporated in verysmall concentrations in one-component phospholipid bilayers differs by several or-ders of magnitude above and below the main phase transition temperature, Tm ofthe host bilayer [24]. In the solid (or gel) phase reported values range between


D ≈ 10−11 cm2/s [30] and D ≈ 10−16 cm2/s [130]. Fluid phases are characterizedby very different transport properties, with D ≈ 10−8–10−7 cm2/s [24, 161].

The diffusion coefficients of integral membrane proteins in fluid lipid bilayersdepend on the size (radius) of the protein in fairly good agreement with the Saffmanand Delbruck theory (eq. (6)). In table 1 we present the experimental and theoreticaldiffusion coefficients for some proteins covering almost the entire interval of possiblesizes. A complication of these experiments is the spontaneous self aggregationof proteins in the reconstituted systems thus leading to a large uncertainty in theestimated radius of the diffusing particles. This, however, could be dependent uponprotein concentration in the bilayer, a possible reason for the disagreement betweensome experimental and theoretical D values [24].

The diffusion coefficients of lipids in fluid lipid bilayers depend on the host lipidbut are essentially independent of the lipid probe that is used to measure diffu-sion [161]. Moreover, diffusion of a tracer is almost independent of its acyl chainlength [161], in strong disagreement with the Saffman and Delbruck theory but inagreement with the free area model. The best studied distinct lipid fluid phases haveD values that, although of the same order, differ on average by a factor of about2 [7, 114] (but see below for the temperature dependence). These fluid phases,designated by liquid-disordered (`d) and liquid-ordered (ò) [59, 60] occur in phos-phatidylcholine/cholesterol mixtures under certain conditions (see below for moredetails; fig. 4d shows the phase diagram). Some experimental diffusion coefficientsof lipid fluorescent probes in these two phases are given in table 2.

3.1.1. Effect of free volume in the bilayer on diffusionSeveral studies that examined lateral diffusion in phosphatidylcholine/cholesterolbinary mixtures have been published [5, 7, 43, 82, 114, 133]. Vaz et al. [161] haveshown that a free-area model provides a quantitative description of diffusion in the`d (Lα) phase of pure phosphatidylcholines.

It is generally believed that cholesterol causes an increase in the density of lipidbilayers or, in other words, a decrease in free volume [43, 77]. This idea is consistentwith structural studies of the `d and ò phases. The `d phase occurs above theTm of the phospholipid at low cholesterol concentrations; the acyl chains of thephospholipid molecules are in a disordered state (containing a high fraction of gaucheconformers). The ò phase occurs at cholesterol concentrations above ∼ 30 mol%,and the phospholipid acyl chains are mostly in a rigid state (with a high fraction oftrans conformers). Models have been proposed for the molecular arrangements ineach liquid phase: in the `d phase, the cholesterol molecules are believed to spanthe hydrocarbon core of both leaflets of the bilayer; in the ò phase, cholesterol isthought to pack like the phospholipid molecules in each leaflet [119, 120].

Therefore, the `d phase should have a relatively high free volume whereas the ò

phase should have a relatively low one. It thus seems appropriate to use cholesterolas a means of varying free volume in the system and testing the applicability of free-volume theories to lipid diffusion. Almeida et al. [7] performed this experiment in di-myristoylphosphatidylcholine (DMPC)/cholesterol bilayers using FRAP, and showedthat the Macedo–Litovitz eq. (13) provides a convenient means of understanding

328P.F.F.

Alm

eidaand

W.L

.C.

Vaz

Table 1Diffusion coefficients (experimental and theoretical values are indicated) of some integral membrane proteins representative of different possible sizes found

in biological membranes.

Integral Membrane Protein Radius Host lipid Lipid/protein T (◦C) Dexperiment Dtheorya Ref.

(A) bilayer molar ratio (cm2/s) (cm2/s)

M-13 Viral Coat Protein 5.4b DMPC 2000 35 4.8× 10−8 4.3× 10−8 [137]

Glycophorin ?c DMPC 4500 35 1.7× 10−8 4.3× 10−8d [158]

Gramicidin C 8.0e DMPC >100 34 4.3× 10−8 4.0× 10−8 [146]

Bovine Rhodopsin 20 DMPC 3000 36 3.3× 10−8 3.4× 10−8 [159]

Acetylcholine receptor (monomer) 15 Soybean lipids >5000 37 3.3× 10−8 3.6× 10−8 [27]

Acetylcholine receptor (tetramer) 60 Soybean lipids 5000 35 3.1× 10−8 2.6× 10−8 [163]

a Calculated from eq. (6) assuming a membrane viscosity η = 1 poise, a bounding fluid viscosity η1 = 0.01 poise, and a membrane height h = 50 A.b Estimated as the radius of one α-helix.c Uncertain, aggregation of monomers is possible.d Value for monomer.e Taken as that of Gramicidin D given in ref. [14].


Table 2Experimental values of diffusion coefficients of lipid fluorescent probes measured in different ho-mogeneous lipid bilayer systems in the liquid ordered (ò) and liquid disordered (`d) phases, as a

function of temperature. For more complete temperature dependence curves see [7, 8, 161].

Lipid probe Lipid bilayer T (◦C) Phase D (cm2/s) Ref.

NBD-POPE POPC 35 `da 6.1× 10−8 [161]

NBD-DMPE DMPC 34 `d 6.9× 10−8 [161]

46 `d 11.0× 10−8

58 `d 16.2× 10−8

NBD-C10-PE DMPC/cholesterol 34 ò 3.3× 10−8 [7]

70:30 46 òb 8.1× 10−8

58 ?c 14.3× 10−8


60:40 46 ò 7.2× 10−8

58 ò 13.2× 10−8


50:50 46 ò 5.8× 10−8

58 ò 9.7× 10−8

NBD-DLPE DSPC 61 `d 13.2× 10−8 [161]

DSPC/cholesterol 58 ò 6.3× 10−8 [8]

70:30

a More frequently known as Lα.b A small fraction (≈ 10%) of `d phase is present.c Close to the critical point.

quantitatively the relation between the diffusion coefficient of a phospholipid and theconcentration of cholesterol, through the molecular area occupied by this moleculein the plane of the bilayer.

These diffusion experiments [7] agree with previous studies [114] when compar-ison is possible. In general, in the ò phase the diffusion coefficient D of a phos-pholipid decreases with increasing cholesterol concentration. However, for choles-terol concentrations greater than 30 mol%, and between 24 and 34◦C D actuallyincreases with increasing cholesterol content. This is in agreement with the de-crease in the deuterium NMR order parameter observed in going from a 2:1 to a 1:1DMPC/cholesterol mixture [119]. The diffusion coefficient shows a strong non-linearincrease with temperature. The ratio of D in pure DMPC to D in DMPC/cholesterol70:30 varies between 3 at low temperatures to 1 at high temperatures.

In the `d phase, D seems to be independent of cholesterol concentration [5, 7,82, 114]. We have argued that this is either because cholesterol does not reduce freevolume in the pure `d phase or because it changes the free volume distribution. Thelatter possibility is consistent with a free-volume analysis of the results of Shin andFreed [133], which indicate that, in this region of the phase diagram, the diffusioncoefficient of a cholesterol analog changes with cholesterol concentration whereas Dof a phospholipid does not. This is explained if cholesterol occupies preferentially


free volumes smaller than those useful for phospholipid diffusion, but which havenecessarily a size useful for cholesterol diffusion.

The results on phospholipid diffusion can be described [7] by a modified Macedo–Litovitz equation,

D =

(kTat(T )

8m

)1/2

exp(−a∗

af−Ea

kT

)(44)

where m is the mass of the probe molecule (which should be essentially equal tothat of the phosphatidylcholine), at(T ) is the total molecular area per phospholipidmolecule, a∗ is the close-packed cross-sectional area of the phospholipid molecules,which should be approximately the area/molecule in the solid phase (about 45 A2)and af is the free area. In the `d phase af = at(T ) − a∗, and in the ò phaseaf = at(T )− a∗ − nacho, where n is the cholesterol to phospholipid ratio and acho isthe area occupied by a cholesterol molecule.

Since at(T ) can be obtained from specific volume and bilayer thickness data, thisequation has the advantage that only the activation energy Ea needs to be estimatedfrom a fit to the experimental data, for pure phosphatidylcholines. In the case ofcholesterol-containing mixtures in the ò phase, the molecular area of cholesterol isan additional parameter, for which a value acho ≈ 27 A2 seems appropriate [7].

The activation energies obtained using (44) are necessarily lower than those ob-tained with an Arrhenius type of equation because part of the temperature dependenceis attributed in the former case to the free area change with T . The values obtainedfor Ea are 2.7, 1.9, 2.1, and 2.5 Kcal/mol for pure DMPC, and DMPC/cholesterol70:30, 60:40, and 50:50, respectively [7]. This equation corresponds essentially toassuming a form f ∝ exp(Ea/RT ) for the frictional factor used previously [161].Whether or not this is the best option is an open question. However, one of the maincriticisms raised against free-volume models – the use of an exaggerated number offree parameters – has been removed, for it is apparent that one or two parametersare sufficient.

3.1.2. Effect of bounding fluid on diffusionVaz et al. [164] have studied the effect of bounding fluid viscosity on lipid diffusionin the bilayer. They used a N -(7-nitro-2,1,3-benzoxadiazol-4-yl) membrane-span-ning phosphatidylethanolamine (NBD-msPE) [162] incorporated into 1-palmitoyl-2-oleoyl phosphatidylcholine (POPC) bilayers and studied its diffusion by FRAP. Thebounding fluid was either water or glycerol, the viscosities of which differ by atleast two orders of magnitude at the same temperatures. The diffusion coefficient ofNBD-msPE was found to differ by almost two orders of magnitude, being approxi-mately 10−8–10−7 cm2/s for bilayers in water and 10−10–10−9 cm2/s for bilayers inglycerol [164].

The temperature dependence of the diffusion coefficient was not adequately de-scribed by continuum hydrodynamic theories of membrane diffusion [56, 117], whenthe appropriate bounding fluid viscosity (η1, theory section) was inserted for the case


of glycerol, confirming previous work that showed that the model was unable to cor-rectly describe lipid diffusion in bilayers formed in water [161].

However, using free volume theory, and more specifically the form of the equationfor lipid diffusion proposed before by Vaz et al. [161], these authors were ableto describe the experimental data quite well for both bounding fluids (water andglycerol) [164]. In that equation the pre-exponential factor is of the form kT/f ,where f is the frictional factor. The analysis of the results suggested that f is relatedto the viscosity of the bounding fluid in some simple way [164]. If one uses theMacedo–Litovitz equation, then this friction is incorporated in the activation energyfor diffusion (cf. above), and the result basically says that the viscosity of the mediumsurrounding the membrane has a large influence on Ea.

3.2. Diffusion in two-fluid bilayers

In order to study diffusion in two-phase systems quantitatively, a knowledge of thephase diagram is essential. Binary mixtures of phosphatidylcholines and cholesterolhave received much attention in this respect recently. Essentially complete phasediagrams have become available for DMPC/cholesterol and dipalmitoylphosphatidyl-choline (DPPC)/cholesterol systems, both from theoretical [59, 60] and experimentalwork using techniques such as NMR, ESR, and differential scanning calorimetry(DSC) [118–120, 170]. These studies have shown that phosphatidylcholine/chol-esterol mixtures constitute monotectic systems, thus confirming indications fromearly studies [38, 83, 112, 132]. The most salient features of this type of system [49]are the existence of a miscibility gap where two fluid phases coexist, and a criticalpoint in the fluid state of these mixtures. Above the Tm of the phospholipid and belowthe critical point, the liquid-disordered (`d) and the liquid-ordered (ò) phases cancoexist at the same temperature and pressure, for cholesterol concentrations between5–20 and 30 mol%.

Phospholipid lateral diffusion in DMPC/cholesterol bilayers in the `d–ò coex-istence region is correctly described by current theories of transport properties ofstatistically isotropic, two-phase systems in two dimensions [65, 66, 91, 149], takinginto account the phase diagram. The diffusion coefficient in the two-fluid regionis correctly given by eq. (19), taking `d to be phase 1, and ò to be phase 2(D`d > Dò). The agreement was very good for DMPC/cholesterol mixtures 85:15,80:20 and 75:25 [7].

That study [7] indicated that it is not possible to determine the line of connec-tivity in the `d + ò two-phase region from this type of analysis, due to the smalldifference between the diffusion coefficients in the two phases. However, Smithet al. [138] studied the lateral diffusion of a fluorescein-labeled integral membraneprotein (M-13 Coat Protein) in DMPC/cholesterol multibilayers at a few tempera-tures, using periodic pattern photobleaching, and noticed that above the Tm of DMPCthe apparent diffusion coefficient has a minimum at ' 20 mol% cholesterol at 26◦C,and at ' 25 mol% cholesterol at 32◦C. Almeida et al. [7] have interpreted that ob-servation as indicating that the minimum corresponds to the percolation threshold inthe `d–ò coexistence region. This suggestion is based on the observation that the


percolation point is often associated with a maximum in the FRAP recovery time,or a minimum in the apparent diffusion coefficient [6, 8]. This behavior would beexpected if M-13 Coat Protein partitions differentially between the `d and ò phases.Thus, it may be possible to estimate phase connectivities in the miscibility gap fromstudies of protein diffusion in this region.

3.3. Restricted diffusion of lipids

3.3.1. Problems with available theoriesRestricted diffusion in solid–fluid, two-dimensional systems, or diffusion in an archi-pelago, has received considerable attention from the theoretical point of view, asdiscussed above. The problem, which is to understand the effect of impermeableobstacles (islands) on the diffusion of a tracer, free to move in the rest of the plane(the sea), was discussed by Saxton [122]. At that time, only two experimental studies,in which integral proteins were shown to restrict lipid diffusion, were available onthe subject [106, 146]. The agreement between theory and experiment was foundto be poor. Monte Carlo simulations underestimate the effect of obstacles on tracerdiffusion, predicting lipid diffusion coefficients that are significantly larger than thoseobserved experimentally.

One explanation proposed for this discrepancy is that a layer of lipid is permanentlybound to the protein molecule, thus increasing the effective size of the obstacles.However, this correction is insufficient to explain the discrepancy [122]. Anotherpossible reason would be an increase in viscosity of the bilayer because of thepresence of proteins. According to Einstein’s theory [33–35], the dispersed particlesincrease the viscosity of a fluid by reducing the transfer of momentum between‘layers’ of fluid [76]; there is no dependence on the size of the particles, but onlyon their total volume. However, this is not consistent with experimental studies (seebelow).

Some authors [6, 122, 143] have suggested or argued that the solid obstacles (lipidsolid phases or proteins) can reduce the diffusion coefficient of the tracer both byhard core repulsion and by increasing the order of the fluid-phase lipids in a boundarylayer around the obstacles. Theoretical studies [1, 61, 86, 102, 103, 143] have treatedan integral membrane protein as an impermeable cylinder embedded in the bilayer,which influences the orientational order parameter of adjacent lipid molecules. Someexperimental support exists for these models [6, 85, 157], and it has been suggestedthat they should be applicable to the interface between lipid fluid and solid phasesas well [6, 172]. The lipid order parameter ψ(r) decays exponentially away from anobstacle [102],

ψ(r) = ψf + (ψb − ψf) exp[− (r −R)/ξ

](45)

where R is the radius of the obstacle placed at the origin, r is the radial distance,and ξ is the coherence length of the decay. Jahnig [61] estimated ξ to be about 15 A.The order parameter has the value ψf in the bulk (unperturbed) fluid phase, and thevalue ψb at the phase boundary. The free area per lipid is related to ψ [102] byu(r) = 1− ψ(r), where u(r) = af(r)/af0 , af(r) being the free area at position r, and


af0 the free area in the bulk fluid phase. The boundary layer thus defined extendsover distances larger than just one lipid molecule, in agreement with the results ofMarsh and Watts [85].

In a two-phase system, we can define a normalized diffusion coefficient D∗(r) asthe ratio of the diffusion coefficient at a given position to its value in the bulk fluidphase D0. Given the relation between the D and the free area, this leads to [6]

D∗(r) =D(r)

D0= exp

[(a∗/a0

)(1−

1u(r)

)]. (46)

The ratio a∗/a0 ' 2.5 for saturated diacyl phosphatidylcholines. According to freevolume theory, the diffusion coefficient in a two-phase system is an average over thedifferent regions of the system, D = 〈D∗(r)〉. This leads to [6]

D = 1− c(R+ δ)2

R2+

2c

R2

∫ R+δ

R

D∗(r)r dr + O(c2) (47)

where c is the obstacle area fraction, O(c2) is a set of terms that correct for overlapsinvolving pairs of obstacles at moderate obstacle densities, and δ (δ ≈ 4.5ξ) is alength beyond which the effect of the obstacle is not detectable.

We note that the mobility of the obstacles does not enter this equation. Theapplicability of the model can be assessed by taking the limit δ → 0 in eq. (47), cor-responding to a system where only hard core repulsions exist. The result is D = 1−c,which indicates that the solution including annular regions should be appropriate forsystems for which D = 1− c approximates the solution for the hard core interactionproblem. It turns out that D = 1 − c works reasonably well both for large, almostimmobile lipid solid-phase obstacles and for small, mobile ones because the diffu-sion coefficient of the obstacle decreases as its radius increases. Notice, however,that the relationship between D and 1/R is linear in three dimensions, as given bythe Stokes–Einstein relation [32, 35, 141] but logarithmic in two-dimensions [116],so that this equation can only be valid for obstacle sizes within a certain interval.

In the following sections, we discuss in more detail some very recent experimen-tal studies of restricted diffusion performed in lipid [6, 9] and in protein–lipid [14]model systems. Both types have some positive and negative aspects if we have inmind a comparison with theory. In two-phase lipid systems the area fractions of thetwo coexisting phases can easily be varied by changing the temperature and evalu-ated from DSC with some assumptions regarding the area/molecule in each phase[6, 9]. In FRAP experiments, the fluorescent probe chosen has to partition essentiallyexclusively into the fluid phase when solid and fluid phases coexist in a lipid bilayer.A drawback is that the size and the geometry of the solid-phase domains are notgenerally known, whereas they are more easily estimated for proteins. However, inprotein–lipid systems a new sample is needed for each desired obstacle fraction andevaluating area fractions is less straightforward.


3.3.2. Lipid diffusion in two-phase lipid bilayersRestricted diffusion has been studied in several lipid systems. Above the percolationthreshold (small fraction of solid) a lateral diffusion coefficient D can be calcu-lated from D = ω2/4τ [142]. [Notice that below pc (disconnected fluid phase) thevalues of τ are not related to the diffusion coefficient in a simple way.] The firstsystem we wish to consider here is N -lignoceroyldihydrogalactosylceramide (Lig-GalCer)/DPPC. In the mixture LigGalCer/DPPC 20:80 the percolation threshold ispc = 0.30 [6]. This is close to the value of pc = 0.33 predicted from percolationtheory for random overlapping disks in a continuum [176]. Thus, it appears reason-able to model the solid phase as composed of units that are roughly circular, andto interpret these results using theories that consider circular or hexagonal obsta-cles. The diffusion coefficient decreases linearly with the solid area fraction (fig. 1).This behavior is consistent both with Monte Carlo simulations of restricted diffusion[122, 124] and with the boundary layer model described above.

Using the Monte Carlo simulation results of Saxton [124], Almeida et al. [6] haveestimated the solid phase obstacles to contain about 14 lipid molecules, if they areconsidered immobile. This value is probably incorrect because such small solid

Fig. 1. Relative diffusion coefficient of a phospholipid fluorescent probe in the lipid mixture LigGal-Cer/DPPC as a function of the area fraction of solid phase. The diffusion coefficients were determinedby FRAP, and the data points represent averages of about 4 measurements. The straight line is consis-tent both with Monte Carlo simulations of tracer diffusion in the presence of centrosymmetric immobileobstacles (however, the domain size predicted would be too small; see text) and with the boundary layer

model. Modified from ref. [6], with permission.


obstacles would have a very short lifetime and a large D (comparable to that ofintegral membrane proteins formed by one α-helix [24]). If the obstacles are mobiletheir effect on diffusion of a lipid molecule is smaller [122], and it becomes necessaryto invoke even smaller obstacles explain the magnitude of the reduction observed inthe diffusion coefficient of a lipid.

Interpretation of diffusion results using the boundary layer model [6] leads to avalue of R/ξ ' 8, indicating an average radius R ≈ 100 A for the solid-phasedomains. The same approach was used to study the DMPC/distearoylphosphatidyl-choline (DSPC)/cholesterol system [9]. Mixtures containing an equimolar mixtureof DMPC and DSPC plus varying concentrations of cholesterol were examined us-ing FRAP. In most of these mixtures the percolation threshold occurs at solid areafractions > 0.50, indicating that the solid-phase obstacles are not too asymmetricand that the approach described could be applied. In these mixtures, between 4 and20 mol% cholesterol, the solid phase unit size is little affected by varying cholesterolconcentration, and we found [9] that R/ξ ' 2− 4, or R ≈ 50 A. Above 20 mol%cholesterol the size of the solid-phase domains seems to decrease.

3.3.3. Lipid diffusion in protein–lipid bilayersThe first two studies of the effect of integral proteins on lipid diffusion were that ofPeters and Cherry [106], using Bacteriorhodopsin and that of Tank et al. [146], usingGramicidin. The most recent and extensive work on this problem is that of Blackwelland Whitmarsh [14], who examined the diffusion of the lipid molecule plastoquinone(PQ) in lipid bilayers in the presence of six reconstituted integral membrane proteins(table 3 and fig. 2), using fluorescence quenching techniques. They showed that PQdiffusion is significantly slower than predicted by three theoretical models [100, 108,109, 122], which consider proteins as obstacles for lipid diffusion. Moreover, in theselipid–protein systems a strong dependence of the lipid diffusion coefficient on proteinsize is observed [14], which is inconsistent with a classical viscosity effect [35] of theprotein particles alone. A small protein such as cytochrome f of spinach chloroplasts(R ' 5 A) restricts PQ diffusion much more than larger proteins (R ' 23 A) do, atthe same fractional areas of protein. The dependence of the lipid relative diffusioncoefficient D on the concentration of Bacteriorhodopsin [106] agrees well with thatobtained by Blackwell and Whitmarsh for proteins with a similar radius.

The dynamic boundary layer model provides a possible explanation for theseexperimental results [6] as can be judged from the fits shown in fig. 2. Table 3summarizes the data on protein size, values of D for a protein area fraction c = 0.1,and the corresponding values of ξ. Leaving out Gramicidin all values of ξ are veryclose to each other for these six proteins (ξ = 22± 2 A).

The obstructive effect of Gramicidin on lipid diffusion observed in this work [14]is in good agreement with Saxton’s simulations [122], which take only into accounthard core repulsions. This is not surprising given the structure of Gramicidin, whichis much smaller (molecular weight Mw = 2 kD) than the other integral proteinsstudied (Mw = 33–250 kD) [14]. Analysis of the results with the boundary layermodel leads to ξ ' 1 for Gramicidin, indicating that there is no boundary layer effectwith this peptide.


Fig. 2. Relative diffusion coefficient of plastoquinone in the presence of six different integral membraneproteins, as a function of the fractional area occupied by the proteins. (A) Reaction Centers fromRhodobacter sphaeroides, (B) Cytochrome bc1 from beef heart mitochondria, (C) Cytochrome bf fromspinach chloroplasts, (D) Cytochrome Oxidase from beef heart mitochondria, (E) Cytochrome f fromspinach chloroplasts, and (F) Gramicidin D. The diffusion coefficients were determined with fluorescencequenching techniques by Blackwell and Whitmarsh [14]. The curves represent the boundary layer model,where the coherence length ξ of this layer is the parameter fitted [6]. See also table 3. Taken from

ref. [6], with permission.

3.3.4. Diffusion of membrane-spanning moleculesBefore ending this section we would like to discuss briefly the problems involvedin diffusion of membrane-spanning molecules such as integral membrane proteins,and review some recent experimental results. In two-phase (or multi-phase) lipidsystems, these membrane-spanning molecules feel a much more complicated ‘land-scape’. Their diffusion can be much more restricted than that of molecules spanningonly one monolayer, because they feel obstacles in both monolayers at the sametime.

This problem was examined experimentally in LigGalCer/DPPC 20:80 and 50:50,


Table 3Membrane proteins that have been used as obstacles to the diffusion of lipids. Relative diffusioncoefficients D of plastoquinone [14] or diI [106] (relative to the value in the absence of proteins)are given for a protein area fraction of 0.1. The protein radius and the coherence length ξ of

the boundary layer estimated according to [6] are indicated in each case.

Integral Protein Radius (A) ξ (A) D at c = 0.1a Ref.

Bacteriorhodopsin 18b 17.3 0.47 [106]

Reaction Centersc 24.6d 22.4 0.47 [14]

Cytochrome bc1e 24.0 24.0 0.5 [14]

Cytochrome bff 21.0 20.2 0.3 [14]

Cytochrome oxidaseg 22.2 21.6 0.45 [14]

Cytochrome fh 5.4 24.2 0.15 [14]

Gramicidin D 8.0 0.94 0.9 [14]

a Estimated from the data given in the corresponding references; c is the area fraction ofprotein in the bilayer.b [24].c From Rhodobacter sphaeroides.d Calculated using the cross sectional areas, assumed circular, given in ref. [14].e From beef heart mitochondria.f From spinach chloroplasts.g From beef heart mitochondria.h From spinach chloroplasts.

and in DMPC/DSPC 50:50 [6, 165], by comparing the diffusion of a membra-ne-spanning fluorescent lipid (NBD-msPE) with the diffusion of a ‘normal’ lipidprobe [NBD-POPE, or NBD-dilauroylphosphatidylcholine (DLPE)]. Both in Lig-GalCer/DPPC and in DMPC/DSPC, NBD-msPE diffuses slower than NBD-POPEand NBD-DLPE, by a factor of 2 when the systems are completely in the fluid state.

Two situations concerning the phase distribution across the bilayer were shownto occur [6]. In one case (DMPC/DSPC), in a single bilayer, solid domains in onemonolayer are exactly superimposed upon solid domains in the apposing monolayer:the diffusional paths of a bilayer-spanning and a monolayer-spanning probe are iden-tical. In the other case (LigGalCer/DPPC), spatial distributions of solid domains inthe two apposing monolayers are independent of each other: the membrane-spanningmolecule senses more domains in its path, and its diffusion is therefore more re-stricted. The effective solid area fraction felt by the membrane-spanning moleculein this case is ceff = 1 − p2, where p2 is the probability of having fluid phase inone monolayer superimposed onto fluid phase in the other monolayer, for indepen-dent distributions of domains in the two monolayers (p is the fluid area fractionin each monolayer). In this case a shift in the apparent percolation threshold to-wards lower fractions of solid phase is also noticeable when the tracer used is amembrane-spanning lipid.


3.4. Percolation

If a FRAP experiment is performed in a sample of multi-bilayers of a lipid mixtureat a temperature at which the system is entirely in the fluid state the fluorescencerecovery is always complete [6, 64, 165]. When this sample is cooled down intothe mixed-phase (solid + fluid) region of the system phase diagram, the recoverybecomes fractional (fig. 3), eventually having very small values.

These temperature scans show little or no hysteresis [6, 165, 166], strongly sug-gesting that the system is at equilibrium at each point [165]. It is important to notethat all the temperature change is doing (with respect to the problem under considera-tion) is changing the relative fractions of fluid and solid phases present in the system,according to the corresponding phase diagram. These FRAP measurements allowfor the location of a unique point in the phase diagram where the percent recoverydrops abruptly. This point has been interpreted [6, 9, 18, 165, 166] as identifyingthe percolation threshold in the plane of the lipid bilayer. Frequently, though notalways, the FRAP recovery time reaches a maximum at the same point [6, 8, 9].

The relevance of phase percolation in heterogeneous lipid bilayers was first sug-gested more than 10 years ago [136, 140] as a means of explaining the results of diffu-

Fig. 3. Fractional fluorescence recovery in a FRAP experiment as a function of the solid area fraction,c, in the binary mixture LigGalCer/DPPC 20:80. The points represent averages of 4 measurements andthe error bars give the corresponding standard deviations. The curve is a fit to an equation of the formR(p) = exp(y)/(1 + exp(y)) with y ∝ (pc − p) (see text), which gives for the percolation threshold(inflection point in the curve) of the fluid phase pc = 0.30 (or a threshold cp = 1 − pc = 0.70 for the

solid phase). Taken from ref. [6], with permission.


sion studies [114] in phosphatidylcholine/cholesterol mixtures. After that, however,not much progress occurred in the field until the recent study of the DMPC/DSPCsystem using FRAP [165]. Since then, several lipid binary systems have been ex-amined comprising the basic types [20, 49] of phase diagrams (isomorphous [166],monotectic [7, 9], eutectic [18], and peritectic [165]), plus a ternary system [9]. Weshall discuss those results shortly, but first we would like to examine the methodsused in their interpretation.

3.4.1. Interpretation of FRAP experimentsOne important aspect is the estimation of the percolation threshold pc from the vari-ation of the fractional recovery with fluid (or solid) area fractions (or temperature).In the theoretical section p represented the fraction of conducting area and c that ofblocked (or empty) area. Here p is the fractional area of fluid phase and c that of thesolid phase. Percolation is used here to mean percolation of the fluid phase. Thus,above the percolation threshold means p > pc, and below the percolation thresholdmeans p < pc. Now, percolation arguments can simply be applied to lipid bilayerswhere fluid and solid phases coexist. The fluid phase is where diffusion occurscorresponding to the conducting phase in percolation theory.

In the first investigations the percolation threshold (also called the point of con-nectivity) was operationally defined as that point at with the recovery ceased to becomplete [18, 165, 166], when the temperature was decreased from an all-fluid stateinto the coexistence region of the phase diagram. More recently, Almeida et al. [9]have identified pc with the point of inflection in the percolation probability of thesystem, R(p) (34). As noted above, several different expressions have been usedfor R(p) in the percolation literature, but the estimation of pc is little sensitive tothe particular form of R(p). Since the samples used in FRAP consist of stacks ofa few hundred bilayers, we have assumed [9] that the fractional recovery gives theprobability of percolation, and used the approximation R(p) = exp(y)/(1 + exp(y)),where y = AL1/ν(p − pc) and the constant A = 3.5 [9]. The boundary conditions(section 2.2.5) are approximately satisfied for large AL1/ν (they are satisfied in thelimit L → ∞). A fit of this equation to experimental results is shown in fig. 3.However, we stress that this form of R(p), though useful to estimate pc and L, is notbased on rigorous theoretical arguments.

The system size L is essentially proportional to the steepness of the recovery droparound the threshold (L1/ν ∝ dR(p)/dp). The system size is well defined in latticeand continuum Monte Carlo simulations of percolation. In the former case L is thelength of the lattice in units of the lattice distance λ; in the latter, it is the lengthof the system in units of the diameter φ of the disks of conducting phase. We havechosen [9] to define L in a two-phase lipid bilayer by L = ω/φ, where ω is the radiusof the bleached spot in a FRAP experiment and φ is some characteristic length ofthe fluid-phase domains. Thus, varying ω and calculating L from R(p) allows for anestimate of φ experimentally. This has not been done so far.

The meaning of φ in two-phase lipid bilayers deserves some comment. The sim-plest interpretation is that it corresponds to the length of some characteristic ‘unitcluster’ or domain. It contains information about the size and the structure of fluid


domains. Subject to some corrections depending on the averaging process [144], webasically have φ ∼ S1/df , thus giving different dependencies of φ on p for differentdimensionalities (df) of the fluid domains.

Finally, we would like to consider the effect of dynamics on percolation. Strictly,a percolation threshold can be observed only if the blocked sites or the obstacles areimmobile [52, 122, 126]. In practice, it is only necessary that the time-scale of thethe solid domain mobility and their lifetimes be much larger than the recovery timeof a FRAP experiment. Both solid domain mobility and fluctuations in time affectpc in the same direction: pc is shifted to lower p, that is, to larger fractions of solidphase, as compared to the equivalent static system. This is because the dynamicperturbations allow the fluid-phase domains to communicate within the period of theexperiment, thus leading to a higher effective connectivity of the fluid domains. Inthe extreme case where fluctuations or solid domain movement are very fast, thethreshold occurs exactly at the onset of the solid–fluid phase transition. This effectcan also be observed if fluctuations are very extensive, which seems to be the casein single-component phospholipid bilayers [166]: a fluorescent probe trapped in adisconnected fluid domain at a given moment can find itself connected to the rest ofthe fluid phase in the subsequent instant because the solid barriers have melted in themeantime. The fluid phase will be perceived to percolate when a fluid cluster spansthe field of the FRAP spot in any direction, whereas percolation theory considersnormally clusters that span the system either sidewards or upwards (or averagesbetween the two). In an infinite system this does not represent a problem, but infinite systems, if pc is taken to correspond to percolation either sidewards or upwards,the threshold is smaller than the true one [178]. Thus, in general, it is likely that thetrue (static, or ‘frozen’) pc be larger than that estimated from FRAP experiments.

3.4.2. Phase percolation in lipid mixturesPhase percolation has been studied in several different lipid systems corresponding todifferent types of phase diagrams (fig. 4). A recent article by Vaz [168] reviews mostof this work. The simplest system is one in which the components are miscible in allproportions both in the fluid and solid phases. This is the isomorphous system andthe example studied was DMPC/DPPC [166]. Two types of systems exhibit completemiscibility in the fluid phase but show miscibility gaps in the solid state. These arethe peritectic and the eutectic cases, which are classified according to the reactionoccurring (as the temperature is raised) at the characteristic point [49]: solid1 →solid2 + fluid in the peritectic case; solid1 + solid2 → fluid, in the eutectic case. Theperitectic system examined was DMPC/DSPC [165], and the eutectic one was di-heptadecanoylphosphatidylcholine (di-C17PC)/ 1-docosanoyl-2-dodecanoylphospha-tidylcholine (C22C12PC) [18]. The monotectic system is characterized by the reactionfluid1 + solid → fluid2; the phase diagram shows a fluid-fluid miscibility gap anda critical point. The examples of this type studied were DMPC/cholesterol andDSPC/cholesterol. Almeida et al. [6] also examined the LigGalCer/DPPC system,the nature of which is not quite clear, probably having a more complex phase diagram(most likely a combination of eutectic or peritectic with monotectic, judging from refs[6, 45]. The ternary system studied was DMPC/DSPC/cholesterol, but restricted to a


Table 4Percolation thresholds (pc) and corresponding temperatures (in ◦Celsius) for the lipid systemsstudied with FRAP. Tpc is generally estimated from the FRAP fractional recovery; the valuesin parenthesis are estimated from the maximum or inflection in the FRAP τ . The valuesof pc are given in area fractions (of fluid), and were calculated assuming molecular areasof 45, 60, and 65 A2 for the solid, liquid-disordered and liquid-ordered phases, respectively.Previously [9] we used a value of 80 A2 for the liquid-ordered phase, corresponding to 50:50phosphatidylcholine/cholesterol mixtures, but the value used here, corresponding to 70:30,appears more correct. The change in the values of pc is however very small (0.03–0.04)for all concentrations. The fractions of the solid and fluid phases are calculated from thephase diagram (DMPC/DPPC, DMPC/DSPC, DMPC/cholesterol, DSPC/cholesterol, and di-C17PC/C22C12PC) or from integration of excess heat capacity curves (DMPC/DSPC/cholesteroland LigGalCer/DPPC). The percolation threshold is taken as the inflection point in the FRAPfractional recovery as a function of fluid (solid) area fraction (an equation of the form R(p) =exp(y)/(1 + exp(y)) with y ∝ (p− pc), see text) [9], or as the point at which recovery ceasesto be complete, in which case the values are marked with an asterisk (*). The length φ of fluid

domains is also indicated for a few mixtures.

SYSTEM (MIXTURES) Tpc pc φ (nm) Ref.

DMPC/DPPC [166]

75:25 27.5 0.52∗

50:50 32.0 0.52∗

25:75 36.5 0.49∗

DMPC/DSPC [165]

65:35 36.0 0.77∗

50:50 43.0 0.73∗

35:65 47.0 0.65∗

20:80 50.0 0.42∗

DMPC/cholesterol [9]

85:15 23.6 (23) 0.42∗

80:20 23.5 (18) 0.65∗

75:25 22.5 (14) 0.85∗

DSPC/cholesterol [9]

85:15 54.0 0.44∗

80:20 48.0 0.66∗

75:25 42.0 0.88∗

DMPC/DSPC/cholesterol [9]

48:48:4 38.0 0.74 430

46.5:46.5:7 39.0 0.71 210

45:45:10 37.0 0.63 130

42.5:42.5:15 32.0 0.45 87

41:41:18 26.0 0.38 63

40:40:20 20.0 0.29 58


Table 4 (continued)

SYSTEM (MIXTURES) Tpc pc φ (nm) Ref.

di-C17PC/C22C12PC [18]

10:90 36.9 a

17.6:82.4 37.0

25:75 36.8

40:60 36.6

60:40 40.0 0.79∗

70:30 43.0 0.62∗

80:20 44.2 0.57∗

90:10 47.0 0.47∗

LigGalCer/DPPC [6]

20:80 42.0 0.30 500

33:67 42.0 0.26

50:50 46.0 0.25

a Up to 40 mol% of di-C17PC the threshold occurs right at the onset of solid–fluid transitionand it is not possible to estimate the phase fractions from the phase diagram.

slice in the three-dimensional phase diagram corresponding to equal concentrationsof DMPC and DSPC.

The percolation thresholds for the different systems and mixtures studied are givenin table 4 and marked in fig. 4. It is apparent that pc depends strongly on the particularsystem. When the mixing is close to ideal (isomorphous system, DMPC/DPPC) pc isclose to 0.5 [166], whereas non-ideality of mixing seems to shift pc to either highervalues (peritectic system, DMPC/DSPC [165], and eutectic system, di-C17PC richarea of di-C17PC/C22C12PC [18]) or lower values (LigGalCer/DPPC [6] and eutecticsystem, C22C12PC rich area of di-C17PC/C22C12PC [18]). In the monotectic systems,DSPC/cholesterol and DMPC/cholesterol [9, 114], at low cholesterol concentrationsthe percolation threshold occurs at similar fractions of the solid and fluid phases,but shifts to higher values of fluid phase as cholesterol concentration increases. Inthe ternary system DMPC/DSPC/cholesterol (mixtures 1 : 1 : x), the percolationthreshold varies considerably depending on cholesterol concentration (x): when thecholesterol content increases, pc decreases from ' 0.75 at very low cholesterolconcentration to ' 0.3 at 20 mol% cholesterol [9]. This is opposite to the tendencyobserved in the binary phosphatidylcholine/cholesterol systems, and it is not clearwhy this difference arises. More investigations are needed for understanding thephysical reasons that determine the location of the threshold in the phase diagram.

Interpretation of the results in these systems with continuum percolation theoryindicates that the shape of solid phase clusters in lipid mixtures varies from almostcircular forms (pc ' 0.3) to very elongated forms (pc ' 0.8). Saxton [127] discussedthe observation of very large values of pc (small fractions of solid at the threshold)and concluded that either the solid phase domains are almost needlelike or theirgeometry is complex. One possible problem also indicated is that if the obstaclesform aggregates of certain types the percolation threshold is not well defined. One


Fig. 4. Percolation thresholds in several systems corresponding to the fundamental types of phase dia-grams of binary mixtures. The percolation thresholds (pc) are indicated on the temperature–compositionphase diagrams for (A) the isomorphous case, DMPC/DPPC [166], (B) the eutectic case, di-C17PC/C22C12PC [18], (C) the peritectic case, DMPC/DSPC [165], and (D) the monotectic case, DMPC/Chole-sterol [7, 9]. Fluid and solid phases are indicated by ‘f’ and ‘s’, whereas suffixes identify different typesof fluids or solids. In the fluid–solid (f + s) coexistence regions, pc are estimated from the fractionalrecovery in FRAP experiments. In the f1 + f2 region in the monotectic system (D), the points representminima detected in the diffusion coefficient of a reconstituted protein [138] which we have interpreted

as an indication of the proximity of pc [7]. Taken from ref. [169], with permission.

way to verify if this is the case in any of the systems considered would be to performthe FRAP experiments using different laser spot sizes. A true pc is independent ofthe size of the system and therefore independent of the laser spot size in a FRAPexperiment.

The characteristic length φ of the fluid phase domains has been estimated tobe 50–500 nm [8, 9]. The calculated values of φ in LigGalCer/DPPC 20:80 andDMPC/DSPC/cholesterol mixtures are also included in table 4. These values arereasonably close to previous estimates based on other experimental techniques [13,


41, 58, 177], for different systems.

4. Experiments in biological systems

4.1. New techniques: single particle tracking

The technique of single particle tracking (SPT) recently developed made it possible(in principle) to study the movement of individual molecules in membranes, by theuse of microscopy combined with labelling procedures [28, 46–48, 131]. A mem-brane protein or lipid molecule is labeled with a fluorescent probe or with colloidalgold particles (attached to the target molecule via ligands such as antibodies), and themovement of the label is followed using fluorescence video microscopy (fluorescentprobes) or video enhanced contrast microscopy (gold particles). The size of the goldparticles is typically about 40 nm, and the movement is followed for about 2–10 s,recording positions every 300 µs – 1 s.

Qian et al. [111] presented a detailed comparison of the methods of FRAP andSPT, with particular emphasis on spatial resolution and statistical problems involvedin each technique. Here, we shall only stress some important conceptual questions.Briefly, FRAP measures diffusion over large distances and provides an ‘automatic’ensemble average of the movement of fluorescently labelled molecules, which (forcomplete recoveries) is directly related to the diffusion coefficient through eq. (1)which in this case is often written as ω2 = 4Dτ , where ω is the radius of the bleachedspot (ω ' 1–5µm) and τ is the experimental recovery time (see ref. [64] for moredetails on FRAP).

In a SPT experiment one measures the x and y coordinates of the center of aparticle attached to a molecule that moves in the two-dimensional membrane. Butnow the squared displacement of the particle, r2 = x2 +y2, is not related to D at all.The only statistically meaningful parameter is 〈r2〉, where the brackets indicate anaverage over a large number of particles (ensemble average) or over a large numberof sufficiently long trajectories of the same particle (‘infinite time’). Then, eq. (1)applies with t being the time of the experiment. Often enough, in the literature ondiffusion studies using SPT, in has not been sufficiently appreciated that in order toobtain 〈r2〉 a very large number of r2 has to be collected. This has frequently led toerroneous conclusions.

One potentially important use of SPT is the detection in cell membranes of move-ments other than free diffusion, such as flow or hindered diffusion. Still, it mustbe kept in mind that the trajectory of one particle is not meaningful by itself forthis purpose, but a large number of particle trajectories must be recorded, and ana-lyzed according to statistical criteria [111, 128]. Individual trajectories that could beinterpreted as indicating trapping within domains or flow occur extremely often insimple random walks [128]. The fact that random walks are not uniformly compactand exhibit an amazing diversity of trajectories for individual particles in the verysame system was recognized long ago by Perrin [105] in his experiments of diffusionof mastic granules in water. However, this has often been ignored by workers inthis field, and only recently a few researchers have pointed out that Monte Carlo


simulations of random walks in two dimensions can be used as a control for theexperimental data [79, 128]. Thus, in order to analyze SPT data, both 〈r2〉 and theprobability of occurrence of individual trajectories must be taken into account [128].

Most experimental work has focused on molecular movement in cell membranesinvestigating the occurrence or not of flow [10, 29, 69, 131], as a function of differentcell processes these movements may be associated with. Here we are only concernedwith SPT as a technique to study diffusion. When the movement of particles isdue to diffusion alone, the diffusion coefficients calculated have generally agreedwith those determined by FRAP both in cells [69, 131] and in model lipid bilayermembranes [79]. At first sight one might be somewhat surprised by this result,because a heavy gold particle of 40 nm could be thought to hinder diffusion ofthe membrane molecule. However, this is not a problem because the diffusioncoefficient D = kT/f is independent of the molecular weight of the particle. Thefrictional factor f depends on particle size [24, 116, 117, 174] but not on particleweight. Also, the bulky gold moiety extending into the aqueous phase is in amedium with a viscosity much smaller than the membrane viscosity, and thus itslarge size does not affect diffusion of the molecule moving in the membrane [24,116, 117, 174] Actually, according to classical theory, the mean square displacement〈r2〉 does depend on the molecular weight [35, 42],

⟨r2⟩ = 4Dt

[1− (mB/t)(1− exp(−t/mB))

](48)

but this dependence would only be noticed for times much shorter than those cor-responding to the actual experiments performed with STP techniques. In a typicalcell, however, the glycocalix presents a rather rigid structure that could hinder dif-fusion substantially through frictional drag exerted on the bulky moiety attached tothe tracer particle.

4.2. Restricted diffusion

4.2.1. Mitochondrial membranesThe diffusion of components of the electron transport chain in the inner mitochondrialmembrane was reviewed by Lenaz [80]. The problem has normally been trying todecide between two possibilities concerning the structure of the these membranes andthe dynamics of their function. In one extreme, the multi-protein redox componentscould be uniformly distributed over the entire membrane and be long-range mobile;in this case the electron transfer process would be controlled by the diffusion ofintermediate carriers, ubiquinone and cytochrome c (cytc), or by the turnovers of theenzymes in the multi-protein components. In the other case the electron transfer chaincomponents would be associated to form still larger complexes of longer or shorterduration. The mechanism of electron transfer would be different in the two cases. Inthe random collision model ubiquinone and cytc diffuse (much faster than the integralproteins) between the multi-protein complexes and transfer electrons upon collision.In the domain model the redox reactions occur within each domain. As pointed outby Lenaz [80], the models would be indistinguishable if the association/dissociation


step of ubiquinone and cytc was faster than the redox reactions. Another questionto be decided is whether or not the whole electron transfer process is diffusioncontrolled. The problem is complicated and a clear cut answer does not seem tobe available, in contrast with the case of photosynthetic membranes to be treatedin the next section. We shall thus focus on the more recent literature and refer theinterested reader to the review of Lenaz [80].

Hackenbrock and collaborators have collected evidence in favor of the randomcollision model recently [21, 22, 51]. Diffusion studies of 3,3’-dioctyldecylindocar-bocyanine (diI) in inner mitochondrial membranes using FRAP [21] suggested thatubiquinone-mediated electron transport is a multicollisional, long-range, obstructeddiffusion process, because factors affecting electron transport also affect lipid diffu-sion. Those authors also concluded that the multi-protein redox components are com-pletely mobile and are not divided up into different pools in the membrane. The dif-fusion coefficients of diI and of Protein Complex III (both D ≈ 10−9–10−8 cm2/s) in-crease with decreasing protein concentration, the proteins diffusing somewhat slowerand being more sensitive to protein content than the lipids.

The apparent activation energies for diI diffusion and electron transport vary in aparallel manner with change in protein concentration [22]. Moreover, the apparentactivation energy associated with ubiquinone diffusion accounts for most of the totalactivation energy in the electron transfer process. However, the interpretation ofthe meaning of activation energies in these complex membranes is far from beingstraightforward [80], and the conclusions may not be that simple.

In another study [51], the diffusion of F1F0-ATP synthase, ADP/ATP translocator,and cytochrome bc1 was studied with FRAP. The collision rates and electron transferefficiencies calculated using the diffusion coefficients determined (D ≈ 10−9 cm2/s)appear to be consistent with the random collision model. The authors argue thatthe differences in D for the three proteins are significant and suggest that theseproteins move independently in the mitochondrial membrane. However, the standarddeviations are very large and the ranges of these D overlap considerably.

In conclusion it seems that the random collision model for the electron transfer inthe inner mitochondrial membrane is becoming generally accepted, where ubiquinoneand cytc would carry electrons between the different protein complexes. The questionof whether or not this process is diffusion-limited overall is still open. The fact thatone step in the transfer chain is diffusion-limited (such as the duroquinol oxidaseactivity, which appears to be limited by diffusion of cytc to cytochrome oxidase)does not necessarily mean that the process is globally diffusion controlled [80].

4.2.2. Photosynthetic membranesDiffusion of both proteins and lipids in photosynthetic membranes is restricted bythe structural organization of the membrane components. This is clearly shown in arecent article by Lavergne and Joliot [74], which describes the principles involvedand the rationale for their application to experimental systems. The electron transferchains in photosynthetic membranes operate with two types of processes. One isthe rapid transfer of electrons (mostly tunneling) between chromophores that arerigidly located within integral membrane proteins (reaction centers). These form


generally large complexes which are immobile on the time scale of electron transfer.The second process is the shuttling of electrons between these complexes, whichis accomplished by small carriers that are either water-soluble (plastocyanin) ormembrane-soluble (plastoquinone).

It has been shown in three independent experimental systems that the differentredox centers and carriers are not in global equilibrium over the entire membrane [62,63, 73–75]. The systems studied contained primary and secondary electron donorsand acceptors, the electron transport chain being blocked by the use of an appropriateinhibitor. The global equilibrium state, that is, the equilibrium concentrations ofthe different redox forms of each member of the electron transport chain, can becalculated from a knowledge of the corresponding mid-point redox potentials. Inthe cases studied the equilibrium constant is of the order of 50, indicating that theprimary donor would be almost completely reduced in all circumstances. However,it was found that the measured redox states of these electron chain members infunctional systems correspond to apparent equilibrium constants very different fromthe calculated value. Under conditions of continuous weak illumination this constantwas found to be of the order of 5.

The explanation presented by Joliot, Lavergne, and collaborators is that the pri-mary and secondary donors are organized in small domains or complexes; a localequilibrium within these domains is attained (in a millisecond or sub-millisecondtime scale) but not a global equilibrium (about 10 seconds are needed to reach it).Since diffusion of the inter-domain electron carriers is restricted, global equilibriumis not achieved in the time-scale of the experiment. The details of domain structureand communication vary among the systems studied, but the fundamental principleis the same: it is the small size of the domains or super-complexes that leads to aquasi-equilibrium poise that differs from a global equilibrium.

A test of the predictions of the effect of compartmentation on the equilibrium poisewas performed using isolated reactions centers of Rhodopseudomonas viridis [73].The electron donor chain consists in this case of a primary donor, P960 (bacteriochlo-rophyll-b), and 4 hemes in cytochrome subunits (two pairs of cyt556 and cyt559). Thereaction centers isolated were not allowed to equilibrate globally because of the lackof redox mediators that would carry electrons between the different complexes. Aplot of the photo-oxidation of cyt556 + cyt559 against that of P960 indicates that alocal equilibrium is achieved, but not a global equilibrium.

A first in vivo study was performed in whole cells of Rhodobacter sphaeroi-des [62, 73, 74]. In this system the primary donor is the P870 center, which receiveselectrons from a series of secondary donors: cytc2, followed by cytc1 and an Fe-Scenter, the latter two being part of the cytb−c complex. By monitoring the light-induced oxidation of these redox components, it was possible to establish that they areorganized in super-complexes of the types (P870)2(cytc2)(cytb−c) and (P870)2(cytc2).Simulation of the levels of % P870reduced against % cytreduced using the stoichiometriesindicated is in excellent agreement with the experimental data, whereas a globalequilibrium curve is not.

A second biological system examined was the chloroplast of green plants [63,74, 75]. In this case, electrons are carried from a quinone (QA) in the photosystem II


reaction center (PS II) to cytb6−f , and, eventually, to photosystem I (PS I), probablyvia the water-soluble carrier plastocyanin [63, 74], which is limited to and diffuseswithin the thylakoid lumen, but this step can be essentially blocked by use of aninhibitor. In the chloroplast the different photosystem centers are already separatedfrom each other on a larger scale: PS II is found only in the appressed regions(granal stacks) whereas PS I is found only in the non-appressed regions (stromallamellae). The distance between these two regions is of the order of 0.1 µm. Theintermediate cytb6−f is found in both areas.

The communication between PS II and cytb6−f is accomplished by the lipid-solublemolecules of plastoquinone (PQ), which diffuse in the membrane. The equilibriumconstant for the redox reactions involving the PS II reaction center-bound QA and thePQ was again determined from the mid-point redox potentials of these chromophores,and it was found to be about 70 in favor of PQ reduction. However, the experimentalapparent equilibrium constant was much smaller, of the order of 4 [63, 74, 75].Additional experiments have indicated that a rigid super-complex structure is not anadequate explanation for the observations in this case. What appears to happen isthat the PS II centers and the plastoquinones are organized in small domains thecomposition of which varies within a certain range. These domains contain a smallnumber of PS II centers (about 2–4 on average [75]) and of PQ (about 7 PQ/PS II).With such small numbers the ratio of PQ/PS II follows a Poisson distribution, whichleads to large structural fluctuations in the composition of the domains. Experimentalconditions of weak continuous illumination will superimpose on this another Poissondistribution of photon absorption by the domains. It is the averaging of these localdistributions of small numbers over the entire membrane that leads to an ensembleapparent equilibrium that differs from a situation of true global equilibrium.

Lavergne et al. [75] have interpreted the results using statistical and percolationarguments. Their theoretical results are similar in many respects to the treatment ofthe effect of domain disconnection on the equilibrium poise of enzymatic reactionsin two-phase lipid bilayers recently published [90, 148].

One intriguing aspect of the work in chloroplasts is the very small number of PSII centers per domain proposed. In a situation of crowding these few proteins couldeffectively trap the small plastoquinones; however, the integral protein area fractionis only about 0.5 [75]. This value could be at or above the percolation threshold forthe proteins, but even above the threshold the pockets of protein-free areas left arelikely to be bound by an average number of proteins much larger that 4. We wouldlike to suggest that part of the reason for PQ restricted diffusion might be the lipidstructure of the thylakoid membrane. The domains containing PS II and PQ wouldhave a lipid composition different from the composition of some other (minor) phaseexisting in-between these domains and preventing communication by restricting thediffusion of PQ, which would have (if this hypothesis is correct) a much smallerdiffusion coefficient in this interstitial phase. Recall that the lipid composition of thethylakoid membrane is very peculiar, with a high level of galactolipids [171], someof which have a very high main transition temperature.


5. Concluding remarks

The physics of diffusion in lipid bilayers, and consequently in biological membranesof which they are the basic structural element, is evidently considerably more com-plex than had been assumed to be the case upto about a decade ago. Particularly, thephase complexity of membranes, with all the related problems for diffusion discussedin this chapter, although recognized since at least the early 1970’s, has been ignoredin the literature until about five years ago. As has frequently been the case in thepast, it was studies on simple model systems that alerted the scientific community tothese, of themselves rather elementary, complexities. Studies that are now undergo-ing in this area are of necessity somewhat simplifying in their approach, but providesome interesting and important insights into the stuctural and dynamic existenceof biological membranes. It is even possible that the existing concept of the fluidmosaic membrane [135] could require some significant reformulation [169]. Con-siderably more work both from the physical-chemical and from the cell-biologicalperspectives, preferably of a convergent nature, will be necessary before the finalword is spoken on this highly interesting theme.

Abbreviations

cyt, cytochromeC22C12-PC, 1-docosanoyl, 2-dodecanoylphosphatidylcholinedi-C17-PC, diheptadecanoylphosphatidylcholinediI, 3,3’-dioctyldecylindocarbocyanineDMPC, dimyristoylphosphatidylcholineDPPC, dipalmitoylphosphatidylcholineDSPC, distearoylphosphatidylcholineLigGalCer, N -lignoceroyl-dihydrogalactosylceramideNBD-C10-PE, N -(7-nitrobenzoxa-2,3-diazol-4-yl)-didecanoyl phosphatidylethanol-

amineNBD-DLPE, NBD-didodecanoylphosphatidylethanolamineNBD-DMPE, NBD-dimyristoylphosphatidylethanolamineNBD-POPE, NBD-1-palmitoyl, 2-oleoyl-phosphatidylethanolamineNBD-msPE, NBD-membrane-spanning-phosphatidylethanolaminePQ, plastoquinonePS, photosystemQA, bound quinone

a, area per lipidaf , free are per lipida∗, critical free area for displacementc, area fraction of obstacles, solid (insulating) phase, or blocked (empty) sitesγ, relative mobility of tracer to obstacle, Euler’s constant (in continuum hydrody-

namics), or overlap parameter (in free volume theory)


d, Euclidean dimensiondf , dimensionality (fractal dimension)D, diffusion coefficientD0, diffusion coefficient in the absence of obstaclesD ≡ D/D0, average relative diffusion coefficientη, viscosity of the bilayerη1, η2, viscosities of the bounding fluidsf , frictional coefficientfi, fraction of phase iφ, a characteristic length of phase domainsk, Boltzmann’s constantò, liquid ordered phase`d, liquid disordered phasep, area fraction of fluid, or conducting, phasepc , percolation threshold (of fluid, or conducting, phase)R, a radius (of domains or proteins)r, distancer0, tracer radiusρ, number densityS, domain size (area)T , temperatureTm, main transition temperaturet, timeτ , a characteristic timeξ, a correlation or a coherence lengthω, radius of bleached spot in FRAP

DSC, differential scanning calorimetryESR, electron spin resonanceFRAP, fluorescence recovery after photobleachingNMR, nuclear magnetic resonanceSPT, single particle tracking

Acknowledgments

This work was supported in part by a grant from the Portuguese Junta Nacional deInvestigacao Cientifica e Tecnologica through the program STRIDE.

We thank Michael Saxton for a preprint of his papers [127, 128], and for manydiscussions on diffusion and percolation, which were an important contribution toimprove our understanding of these problems.

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