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Crowding Effects onDiffusion in Solutionsand CellsJames A. Dix1 and A.S. Verkman2

1Department of Chemistry, State University of New York, Binghamton,New York 139022Departments of Medicine and Physiology, University of California,San Francisco, California 94143; email: Alan.Verkman@ucsf.edu

Annu. Rev. Biophys. 2008. 37:247–63

First published online as a Review in Advance onFebruary 4, 2008

The Annual Review of Biophysics is online atbiophys.annualreviews.org

This article’s doi:10.1146/annurev.biophys.37.032807.125824

Copyright c© 2008 by Annual Reviews.All rights reserved

1936-122X/08/0609-0247$20.00

Key Terms

Brownian diffusion, anomalous diffusion, molecular crowding, cellmembrane, cytoplasm

AbstractWe review the effects of molecular crowding on solute diffusionin solution and in cellular aqueous compartments and membranes.Anomalous diffusion, in which mean squared displacement does notincrease linearly with time, is predicted in simulations of solute dif-fusion in media crowded with fixed or mobile obstacles, or whensolute diffusion is restricted or accelerated by a variety of geometricor active transport processes. Experimental measurements of solutediffusion in solutions and cellular aqueous compartments, however,generally show Brownian diffusion. In cell membranes, there are ex-amples of both Brownian and anomalous diffusion, with the latterlikely produced by lipid-protein and protein-protein interactions.We conclude that the notion of universally anomalous diffusion incells as a consequence of molecular crowding is not correct and thatslowing of diffusion in cells is less marked than has been generallyassumed.

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Crowding: theexclusion of solventvolume arising fromthe presence of alarge number ofsolute particles

Brownian diffusion:diffusion in whichthe MSD of aparticle increaseslinearly with time

Anomalousdiffusion: diffusionin which the MSD ofa particle does notincrease linearly withtime

Contents

INTRODUCTION. . . . . . . . . . . . . . . . . 248PHYSICS OF DIFFUSION . . . . . . . . . 248

Normal Diffusion . . . . . . . . . . . . . . . . 249Anomalous Diffusion . . . . . . . . . . . . . 249Anomalous Diffusion in

Membranes . . . . . . . . . . . . . . . . . . . 250Computational Approaches

to Diffusion . . . . . . . . . . . . . . . . . . . 252EXPERIMENTAL

MEASUREMENTS INSOLUTIONS AND CELLS . . . . . 254Experimental Approaches

and Limitations . . . . . . . . . . . . . . . 254Does Aqueous-Phase Molecular

Crowding Always ProduceAnomalous Diffusion? . . . . . . . . . 255

Is Membrane Protein DiffusionUniversally Anomalous? . . . . . . . 256

Diffusion in Cellular AqueousCompartments . . . . . . . . . . . . . . . . 257

CONCLUSIONSAND PERSPECTIVE . . . . . . . . . . . 259

INTRODUCTION

The cytoplasm and the aqueous compart-ments of intracellular organelles such as mito-chondria are crowded with small solutes, sol-uble macromolecules, skeletal proteins, andmembranes. Cell membranes are crowdedwith lipids, some of which are organized intoraft structures, and proteins, some of whichare tethered to skeletal proteins and containextensive external appendages. The conse-quences of this crowding remain a contro-versial and somewhat confused topic. Popularpictorial representations of the aqueous envi-ronment within cells (16) suggest that crowd-ing would seriously hinder solute diffusion—amajor determinant of metabolism, trans-port phenomena, signaling, and cell motil-ity. One possible consequence of molecu-lar crowding and hindered diffusion is the

need for compartmentalized metabolism toovercome diffusive barriers. A second pre-dicted consequence of molecular crowdingis that the physical chemistry of interac-tions in cells, such as protein-protein asso-ciations and enzyme reactions, is drasticallyaltered.

This review focuses on the consequencesof molecular crowding on translational diffu-sive phenomena in biological systems. Theo-retical considerations and computational dataregarding Brownian and non-Brownian diffu-sion are discussed, and experimental evidenceon diffusion measurements in solutions and incell aqueous compartments and membranesis reviewed. We conclude that crowding incell aqueous compartments is largely Brow-nian with diffusion coefficients less than oneorder of magnitude slower than in water. Wealso conclude that protein diffusion in mem-branes can be Brownian if diffusion is slowedby crowding alone, but that specific interac-tions between proteins and lipids can produceanomalous diffusion.

PHYSICS OF DIFFUSION

Translational diffusion is the movement of asubstance from one region of space to another.In a homogeneous solvent where solute sizeis comparable to or greater than that of thesolvent, solute movement is described well byphenomenological or statistical equations inwhich the primary determinants of diffusionare solute size and shape. We call this typeof diffusion normal diffusion. In inhomoge-neous environments, or where the solute issmaller than the solvent, or where a largefraction of the solution volume is occupiedby another solute (a crowder), more complexequations may be necessary to describe solutemovement. We call this type of diffusion inwhich solute movement is not described bythe equations of normal diffusion as anoma-lous diffusion. In the context of this review,Brownian diffusion and normal diffusion arethe same.

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Normal Diffusion

The flux of solute, J, through a planar areain space is proportional to the concentrationgradient across the plane:

J = −D∇C, 1.

where C is the concentration of solute and theoperator ∇ represents the derivative of C withrespect to spatial coordinates. C is generallya function of both time and space. Requiringmass balance, the diffusion equation becomes

∂C/∂t = ∇ · (D∇C). 2.

Equations 1 and 2 are Fick’s first and sec-ond laws of diffusion, respectively. Solutionsto Fick’s laws are given by Crank (6); addi-tional solutions are given by Hines & Mad-dox (20). Fick’s laws are phenomenologicallaws that describe the spatial and temporaldissipation of a concentration gradient. Thediffusion constant can be concentration de-pendent, although in practice D is assumedto be constant. For constant D, Fick’s secondlaw becomes ∂C/∂t = D∇2C . Diffusion canalso be anisotropic, in which case D becomes atensor.

In the absence of a concentration gradi-ent, normal diffusion is usually described byEinstein’s equations of Brownian motion (11).For a solute comparable to or larger than thesolvent, the diffusion coefficient is given by

D = kT/ f, 3.

where k is Boltzmann’s constant, T is theabsolute temperature, and f is the solventfriction coefficient. In normal diffusion, thesolvent is usually thought of as a continuoushydrodynamic fluid in which the details of thesolvent structure and the solvent-solute inter-action are ignored. For a spherical particle in ahydrodynamic solvent of shear viscosity η, thesolvent friction coefficient in Equation 3 is

f = 6πηrh, 4.

where rh is the hydrodynamic radius of theparticle. Equation 4 corresponds to stickboundary conditions between the solute andsolvent in which the hydrodynamic solvent

Mean squareddisplacement(MSD): the squareof the displacementof a particle at sometime relative to theposition of theparticle at zero time,averaged over manyparticles

is considered to stick to the solute at thesolute-solvent boundary. For slip boundaryconditions (no stickiness between the solventand solute), the factor 6π in Equation 4 isreplaced by 4π . For nonspherical shapes, thefriction coefficient is multiplied by a shapefactor that is greater than 1. From Fick’s lawsor from Einstein’s relations, the mean squareddisplacement (MSD), 〈r2〉, of a solute particlein three dimensions is related to the D by

〈r2〉 = 6Dt. 5.

For one and two dimensions, the factor 6 inEquation 5 is replaced by 2 and 4, respectively.The validity of Equation 5 for solute diffusionin fluid phases has been demonstrated manytimes over the past 100 years (19).

The equations of normal diffusion restupon the central limit theorem: The aver-age displacement of a particle is Gaussian-distributed if the displacements themselveshave a finite second moment (i.e., a finitesquared deviation from the origin) and areMarkovian (i.e., the probability of a particulardisplacement is independent of previous dis-placements) (30). Displacements that do notfollow this central limit theorem, such as dis-placement distributions having long tails thatdo not approach zero exponentially, or dis-placements that are correlated, will not in gen-eral lead to normal diffusion. These processeslead to anomalous diffusion.

Anomalous Diffusion

A central assumption in describing normal so-lute diffusion is that the solute diffuses in acontinuous hydrodynamic fluid. This assump-tion is clearly not valid in most biological sys-tems. For example, cell cytoplasm containsmany different solutes with a large distribu-tion of sizes (26), so that diffusion of any onetype of solute in the cytoplasm would notoccur in a hydrodynamic fluid. Also, thereare boundary effects as the solute encountersspectrin networks or organellar membranes.For these and other reasons, one would notanticipate a priori that solute diffusion in

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biological systems would be described by theequations of normal diffusion. Despite this,solutes in many biological systems do followthe equations of normal diffusion (see Experi-mental Measurements, below). In other cases,solutes do not follow normal diffusion.

Following Bouchaud & Georges (4), wedefine solute diffusion that cannot be de-scribed by Einstein’s equations for Brownianmotion (Equations 3 and 5) as anomalous dif-fusion. The non-Brownian behavior is usuallydescribed by the equation:

〈r2〉 = 6Dtα. 6.

Although strictly true if the solute hops fromone potential trap to another potential trapunder restricted conditions (4), Equation 6has nonetheless been used extensively as asemiempirical description of anomalous dif-fusion. If α < 1, the diffusion is called anoma-lous subdiffusion, and if α > 1, the diffusionis called anomalous superdiffusion.

In noncrowded systems, the solute has al-most the entire hydrodynamic fluid (the sol-vent) in which to diffuse. Crowding is the re-duction of the available solvent volume by acrowder (volume exclusion). The crowder canbe either mobile, as in intracellular globularproteins, or fixed, as in a spectrin network.A consequence of volume exclusion is thatthe effective solute concentration increases,thus increasing the chemical potential of thesolute. This thermodynamic consequence isdiscussed by Minton and colleagues (56) inthis volume. In terms of diffusion, the crow-der provides barriers to solute movement. Ifcrowding gives rise to anomalous diffusion, itwill always be manifest as anomalous subdif-fusion (α < 1 in Equation 6) at long enoughtimes and distances. Figures 1a,b define nor-mal and anomalous diffusion in terms of MSDplots and displacement distributions.

Even if a particle undergoes normaldiffusion, the MSDs may not be linear withtime (Equation 5) for all time and distances.As discussed by Berne & Pecora (3), at veryshort times 〈r2〉 of a Brownian solute isproportional to t2, corresponding to free

motion of a particle under the force ofnearby solvent molecules (ballistic diffusion).At longer times, the particle experiencesthe frictional force of the hydrodynamicsolvent. As formalized by the Langevinequation, 〈r2〉 increases linearly with timein this regime. The crossover from ballisticto Brownian diffusion occurs on a timecharacteristic of the correlation time of thevelocity autocorrelation function. For thecase of distance-limited diffusion, in whichthe particle is constrained to move in a limitedtwo- or three-dimensional area or volume(e.g., solutes constrained to move withinan organelle or a corral on the surface of amembrane), there is a limit to the distancea solute can diffuse, resulting in a plateau in〈r2〉 at long times. Thus, the interpretationof 〈r2〉 that varies linearly or nonlinearly withtime must be tempered with the characteristictime over which the measurement is made.An observation of anomalous diffusion fora solute can therefore be compatible withnormal solute diffusion over biologicallyrelevant timescales, but anomalous diffusionover the measurement timescale. Further, asdiscussed below, there are experimental arti-facts that can give rise to apparent anomalousdiffusion. Because diffusion time and distanceare related (Equations 5 and 6), similarconsiderations apply to biologically relevantand experimentally measured distances.

Anomalous Diffusion in Membranes

The Singer-Nicolson model of biologicalmembranes (43) posits a two-dimensionallipid sheet into which proteins are embed-ded. In a sense, diffusion of lipids and pro-teins in a membrane is a process occurringat nearly maximal crowding, because the bi-ological solvent, water, is at very low con-centration. Alternately, the membrane lipidscan be thought of as the solvent into whichproteins are dissolved. For pure lipid bilayers(no protein or other membrane constituents),diffusion of individual lipids appears to beBrownian (28, 38, 45). However, diffusion of

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a

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MS

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Superdiffusion

Subdiffusion

Normal

Normal

b

Displacement

Anomalous

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Figure 1Characteristics and simulations of anomalous diffusion. (a) MSD curves defining normal (Brownian)diffusion and anomalous subdiffusion (downward curvature) and superdiffusion (upward curvature).(b) Distribution of displacements for normal and anomalous diffusion. Initial particle position is at theorigin. For normal diffusion, the distribution is Gaussian and gives rise to Brownian motion. The curvelabeled anomalous has long tails and an infinite second moment, resulting in nonlinear MSD plots andanomalous diffusion. (c) MSD plots for simulations of crowding in an aqueous phase. Simulations donefor 75-nm-radius spherical particles in a 3 × 3 × 9 μm box for 1 ms using the method of Dix et al. (10).(d) (Top) MSD distributions for simulation of 75-nm-radius particles for 10 ms at 9% volume exclusion.The smooth curve is fitted using the expected distribution for normal diffusion. (Bottom) Differencebetween fitted and observed MSD distributions.

lipids and proteins in plasma membranes ofcells can show anomalous diffusion (14, 24,44) as well as normal diffusion (54). The vari-ation in experimentally measured diffusionproperties in the plasma membrane may re-late, in part, to the timescales over which themeasurements are made (24). The finding of

anomalous diffusion of membrane proteinshas led to an update to the Singer-Nicolsonmodel (52), in which clusters of proteins orlipids, such as rafts, contribute to biologicalfunction.

The presence of fixed barriers to diffu-sion can in effect produce volume exclusion

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on the two-dimensional membrane surface(41). These effects are often framed in per-colation theory (46). In percolation theory, asthe concentration of obstacles increases, so-lutes encounter increasing difficulty in find-ing their way around obstacles until a perco-lation threshold is reached. At the percolationthreshold, the fraction of excluded volume issuch that the solute becomes trapped. Simu-lations of these kinds of barriers reveal initialanomalous diffusion, followed by normal dif-fusion (41). The crossover from anomalousdiffusion to normal diffusion occurs at longertimes as the percolation threshold is reached.For mobile membrane proteins, crowding canhave a significant effect on the mobility ofmembrane constituents in model membranesystems (36, 40).

The Singer-Nicolson membrane modelhas recently been recast to take into accountnew experimental data (12, 24, 52). The up-dated model views the plasma membrane asa dynamic mosaic composed of clumps con-sisting of lipids, proteins, and lipid/proteincomplexes. These clumps are dynamic, re-arranging often in response to the biologicalneeds of the cell. Given this updated mem-brane model, it is not clear whether crowding(as defined by simple volume exclusion) is auseful model to explain mobility in cell plasmamembranes. Two biologically relevant mod-els of membrane mobility are protein bind-ing and unbinding to fixed anchors on themembrane (47), and diffusion within and be-tween membrane microdomains such as lipidrafts (21). Binding must be transient to allowmeasurable mobility. The protein is thoughtto remain fixed at its binding site for somelength of time and then hop to another bind-ing site. This model contrasts with the normaldiffusion model discussed above, in which theparticle incrementally moves a distance takenfrom a Gaussian distribution. In the bindingmodel, for sufficiently small time increments,there is a less than unity probability that a par-ticle will move. With a non-Gaussian distribu-tion of unbinding times, diffusion is anoma-lous (4).

Microdomains, corral proteins, or lipidswithin the membrane can exclude proteinsand lipids located outside the corral. Proteinsor lipids may diffuse normally within the cor-ral as well as hop between inside and outside ofthe corral. The diffusion times in the two envi-ronments may be different. As is the case fortransient binding, such movement can pro-duce anomalous subdiffusion.

As pointed out by Kusumi et al. (24), thesmall reduction in protein diffusion coeffi-cients upon aggregation suggests that proteinaggregation by itself probably does not playa significant role in anomalous diffusion inplasma membranes. For example, using themembrane hydrodynamic model of Saffman& Delbruck (39), the diffusion coefficient fora cylinder of radius r in a two-dimensionalmembrane varies only logarithmically with1/r. A doubling of the radius of the cylin-der decreases the diffusion coefficient by only10%. In contrast, for example, the diffusioncoefficient of bradykinin receptors decreasesby a factor of 10 when the receptor is coupledwith G proteins (37). These considerations in-dicate that specific protein-membrane inter-actions can dominate the translational mobil-ity of proteins on the plasma membrane.

Computational Approachesto Diffusion

Given the complexity of the models used toexplain anomalous diffusion, many papers re-porting experimental studies of anomalousdiffusion also report ancillary computationsto interpret the experimental data. The goalof simulations of diffusion is to tabulate thespatial position of particles as a function oftime (the trajectory). The trajectory is thenprocessed to simulate experimental data (10).

As reviewed by Takahashi et al. (49), thereare several methods to obtain the trajectory.The most detailed trajectory is given by anall-atom molecular dynamics simulation inwhich solute-solute and solute-solvent inter-actions are specified on an atomic level. Un-fortunately, both the size (femtoliter) and time

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(μs) of cytoplasmic or membrane systems tobe simulated are too large to simulate withcurrent computer hardware and software.

A common alternative approach to simu-late crowding is stochastic dynamics simula-tion, in which the particle’s displacement iscalculated from forces from nearby particles aswell as from a random displacement (10, 13).This method assumes that particles have noinertia; they are stationary in one position andthen jump instantaneously to another station-ary position. The solvent is considered a con-tinuous fluid without structure. Typically, in-termolecular interactions are represented byLennard-Jones and Coulomb types of poten-tials. With this method, time steps of adequatesize can be used so that areas and volumes ap-plicable to crowding experiments can be sim-ulated (29).

Figure 1c,d give examples of moleculardynamics simulation with Lennard-Jones re-pulsive potentials (J. Dix & K. Hiranuka,unpublished observations). Figure 1c showsanomalous subdiffusion, seen as downwardcurvature in MSD plots, in simulations withlarge particles at 7 and 9 volume% exclusion.Figure 1d gives the distribution of MSDs forthe simulation at 9 volume% exclusion, com-pared with what would be expected for Brow-nian diffusion. The difference suggests thatthe anomalous diffusion at 9 volume% exclu-sion arises because of a shift in the MSDs to-ward higher values. The shift may occur asthe large particles, trapped in a cage by otherlarge particles, suddenly break free and jumpto another cage, mimicking some aspects of aLevy walk and giving rise to anomalous dif-fusion (4). Smaller particles diffusing in so-lutions containing large crowders find theirway around the crowder particles, resultingin slowed but Brownian diffusion (10).

The simulation time and space regimesneeded for crowding computations are barelyaccessible with common laboratory comput-ers, even at fairly low crowder concentrations.For example, simulation of 1 μm3 volume with25 volume% mobile crowders 10 nm in di-ameter for 1 ms (values typical for simulating

experimental data) requires keeping track of6 × 104 molecules for 1 × 106 time steps; ateach step, costly interaction potentials have tobe calculated. This type of simulation wouldrequire several weeks on a typical fast labora-tory computer. As the volume percent occu-pied by the crowder increases, the number ofparticles increases and the time step decreases(to avoid a particle straying into the excludedvolume of another particle), greatly increasingcomputational time.

A common variation of the stochastic dy-namics simulation method is to specify a lat-tice within the simulation area or volumeand to restrict jumps to vertices of the lat-tice (32, 40), thereby increasing the computa-tional speed by eliminating the computationof interaction potentials. The lattice jumps aregoverned by a set of rules appropriate to thesimulation; typically, jumps are not allowed tovertices already occupied by a molecule. An-other approach is to allow jumps to any regionof space subject to constraints but to ignore in-termolecular interactions (1). Other methodsof trajectory simulation have been developed,such as dissipative particle dynamics, in whichintermolecular forces are computed for parti-cles surrounded by blobs of solvent of vari-able volume (15), and Green’s function reac-tion dynamics, in which particles are moved atvariable time steps and the pair-wise interac-tions are solved exactly by Green’s functions(51). These other methods have not been ap-plied extensively to crowding simulations.

Stochastic simulations of crowding in cy-toplasm reveal both normal and anomalousdiffusion. Dix et al. (10) found that for dif-fusion of 0.7-nm-radius particles, 150-nmcrowders, and a repulsive potential, diffusionwas normal but slowed by a factor of 2.5 at50–60 volume% crowder. Weiss et al. (55)found that in simulations with 3.6- to 5.4-nm particles, a Poisson distribution of largecrowders, and a repulsive potential, diffusionwas anomalous and slowed at 13 volume%crowder. Dix and Kazushi (unpublished ob-servations) found slowed and anomalous dif-fusion at 25 volume% crowders with 150-nm

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Single-particletracking (SPT): anexperimental methodto follow a singlelabeled particle overtime

GFP: greenfluorescent protein

Fluorescencerecovery afterphotobleaching(FRAP): anexperimental methodin which a samplevolume containingfluorophore isbleached and thesubsequent recoveryof fluorescence isfollowed

particles, 150-nm crowders, and a repulsivepotential, as well as smaller-sized particles(Figure 1c,d). Lipkow et al. (25) found atwofold decrease in diffusion of an 8-nm-radius particle at 41 volume% with infinitestep potentials. These simulations indicatethat in all cases crowders reduce particle diffu-sion and that under some conditions diffusioncan become anomalous.

Simulation results on crowding are sub-ject to several caveats. A major assumptionin most simulations is that interparticle in-teractions can be approximated by pair-wiseadditive potentials. At high crowder con-centrations, this assumption becomes lesstenable as three-body and higher-order inter-actions become increasingly important. Ap-parent anomalous behavior seen with two-body interactions may become averaged outwhen higher-order interactions are taken intoaccount. Simulations based on lattice modelsare subject to lattice artifacts when distancesare analyzed on the order of the lattice spac-ing. For crowding at high volume percent,these distances are the predominant distancesto be analyzed. Most crowding simulationsneglect hydrodynamic interactions and weakattractive and long-range potentials. Con-sideration of these interactions may reducethe tails on non-Gaussian displacement dis-tributions, changing the diffusion type fromanomalous to normal.

EXPERIMENTALMEASUREMENTS INSOLUTIONS AND CELLS

Experimental Approachesand Limitations

The continuous, high-resolution tracking ofthe motion of individual molecules in threedimensions is the benchmark in describingdiffusive phenomena. Single-particle tracking(SPT) involves the selective labeling of pro-teins or lipids with fluorophores, such as quan-tum dots, green fluorescent protein (GFP) ororganic dyes (e.g., cyanine dyes), or probes

visible with transmitted light (gold or la-tex beads), such that particle position can bemeasured with as low as nanometer spatialand submillisecond temporal resolution usingsuitable camera detectors. Unlike ensemble-averaged methods to measure diffusion, SPTprovides information about individual parti-cles and so can identify heterogeneous andcomplex diffusive behaviors such as transientconfinement or barriers (22). SPT has becomethe method of choice for studying the two-dimensional diffusion of membrane proteins;however, it is not yet suitable for measurementof diffusion of aqueous-phase solutes in threedimensions because of their generally rapiddiffusion as well as limitations in determina-tion of particle z (axial) position.

Fluorescence recovery after photobleach-ing (FRAP) has been used extensively for dif-fusion measurements. Fluorescently labeledmolecules are introduced into cells by mi-croinjection or incubation or by targetedexpression of fluorescent proteins. In spotphotobleaching, fluorophores in a definedvolume of a fluorescent sample are irreversiblybleached by a brief intense laser pulse. Us-ing an attenuated probe beam, the diffusionof unbleached fluorophores into the bleachedvolume is measured. A variety of optical con-figurations, detection strategies and analysismethods have been used to quantify diffu-sive phenomena in photobleaching measure-ments (reviewed in Reference 53). Recently,we developed a microfiberoptic epifluores-cence photobleaching method in which pho-tobleaching is done at the tip of a micron-sizedfiber that can be introduced deep into solidtissues such as tumors and brain (50).

Besides being an ensemble-averagedmethod describing the averaged diffusive pro-perties of many fluorescent particles, FRAPstudies are generally limited in measure-ment time, such that long-tail phenomenaexpected in anomalous subdiffusion are easilyoverlooked and misinterpreted as incom-plete recovery arising from diffusion-inaccessible compartments (14). As dis-cussed by Periasamy & Verkman (35), the

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determination of solute diffusion coeffi-cients from fluorescence recovery curveshape is challenging when multiple diffus-ing species are present, or when diffusionis anomalous or geometrically restricted.Quantitative comparison of recovery curveshapes measured in cells to that in stan-dards (fluorophore in thin layer of saline)is useful in the experimental determinationof diffusion coefficients (23). Another poten-tial problem in the interpretation of FRAPdata is the presence of reversible photobleach-ing processes, such as triplet-state and flickerphenomenon, which produce recovery signalsunrelated to fluorophore diffusion (34).

Fluorescence correlation spectroscopy(FCS) relies on the analysis of fluctuationsin the number of fluorescent particles in afemtoliter volume defined by a focused laserspot. Increased diffusion results in more rapidfluctuations and a smaller probability that aparticle found in the beam initially will befound in the beam at a later time. This prob-ability is quantified by the autocorrelationfunction. Although FCS methods have beenused to study molecular diffusion in cells(reviewed in Reference 2), it remains uncer-tain given available SPT, FRAP, and directimaging methods whether FCS methods canprovide clear-cut quantitative informationin the complex cellular environment. FCSmethods require very low fluorophore con-centrations and are confounded by reversiblephotophysical processes, cell autofluores-cence, and complexities in beam and cellgeometry. Subtle differences in shape ofautocorrelation functions due to such effectsare easily mistaken for anomalous diffusionwhen fitted to an equation that assumessingle-component isotropic diffusion withideal Gaussian beam geometry and absenceof photophysical phenomena.

From these considerations we concludethat where experimentally possible, as in thecase of diffusion in membranes, SPT is pre-ferred because it provides direct informa-tion about trajectories of individual parti-cles. When SPT is not feasible because of

Fluorescencecorrelationspectroscopy(FCS): anexperimental methodin which fluctuationsin fluorescenceintensity arecorrelated

limitations in labeling or rapid diffusion,as in the case of diffusion in aqueous cel-lular compartments, FRAP provides usefulensemble-averaged information about diffu-sion. FCS is of particular utility in artificialsolutions because of its wide dynamic timerange, though its utility in complex cellu-lar environments may be limited as discussedabove.

Does Aqueous-Phase MolecularCrowding Always ProduceAnomalous Diffusion?

It is widely believed that crowding producesanomalous diffusion. However, we have re-ported FCS measurements in aqueous solu-tions containing diffusing solutes and crow-ders (7), showing that this is not the case atleast to a crowder concentration of 60 vol-ume%. Ficoll 70 was used as the crowderbecause it is noninteracting and of interme-diate size, such that smaller and larger diffus-ing solutes can be studied. Figure 2a showsFCS data for diffusion of the small soluterhodamine green in saline solutions contain-ing the crowder. The data fitted well to asimple Brownian diffusion model for a sin-gle species described by a single correlationtime, τc. In FCS, τc is inversely propor-tional to the diffusion coefficient. As expected,τc increased greatly with Ficoll 70 concen-tration. Anomalous diffusion was not seeneven at the highest crowder concentration.Figure 2b summarizes deduced diffusion co-efficients, showing an ideal, exponential de-pendence of Dw on Ficoll 70 concentra-tion. Diffusion was also measured for a seriesof rhodamine green–labeled macromolecules,including albumin, linear double-strandedDNAs, dextrans, and nanospheres. Remark-ably, although the diffusing substances dif-fered greatly in size, physical properties, andabsolute diffusion coefficients, being bothsmaller and larger than the crowding agent Fi-coll 70, FCS data fitted well to a simple Brow-nian diffusion model with similar exponen-tial dependences of diffusion coefficient on

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crowder concentration (Figure 2c). Of note,hard-sphere models of diffusion in crowdedmedia (18, 31) predict significant deviationsfrom exponentials in the range of Ficoll 70concentrations studied here, a prediction thatwas not confirmed in FCS. Further work isneeded to resolve the apparent discrepancybetween results predicted by simulation andthose obtained experimentally.

Is Membrane Protein DiffusionUniversally Anomalous?

Diffusion in membranes is remarkably slowerthan in aqueous compartments and canbe complex because of membrane crowd-ing with proteins, the presence of distinctlipid domains, and membrane-cytoskeletal in-teractions. As mentioned above, there aremany examples of nonsimple diffusion of

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Time (min)

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Figure 3Long-range nonanomalous diffusion of aquaporin-1 (AQP1) water channels in cell plasma membranes.(a) Schematic of a quantum dot (Qdot)-labeled AQP1 monomer. Labeling was done with an engineeredhuman c-myc epitope inserted between residues T120 and G121, which reside in the second extracellularloop between transmembrane helices M4 and M5. (b) AQP1 trajectories shown over ∼5 min in theplasma membrane COS-7 cell. Each trajectory is shown in a different color. (c) MSD versus time curvesfor AQP1 diffusion in COS-7 and MDCK cells (∼300 individual trajectories averaged for each cell type).Adapted from Reference 5.

membrane proteins, which are generally re-lated to confinement due to protein-proteininteractions. However, it has not been clearwhether anomalous diffusion of membraneproteins, as a consequence of protein/lipidcrowding, is a universal phenomenon. Werecently used SPT to demonstrate long-range, nonanomalous diffusion of aquaporin-1 (AQP1) (5), which is an integral mem-brane protein that facilitates osmotic watertransport across cell plasma membranes inepithelia and endothelia. AQP1 is presentin membranes as a tetrameric associationof monomers, each of molecular mass of∼30 kDa. AQP1 has no known interac-tions with cytoplasmic or membrane pro-teins. We tracked the membrane diffusionof AQP1 molecules labeled with quantumdots bound to an engineered external epi-tope (Figure 3a) at frame rates up to 91 Hzfor more than 5 min (trajectories shown inFigure 3b). In several cell types, nearly lin-ear MSD plots were obtained (Figure 3c)over long and short times. From severaldifferent methods of single trajectory anal-ysis, it was determined that the majorityof AQP1 molecules diffused in a Brownian

manner. Thus, anomalous diffusion of mem-brane proteins is not a universal phenomenon,and when anomalous diffusion occurs, thereshould be an identifiable cause(s) other thansimple crowding by volume exclusion.

Diffusion in Cellular AqueousCompartments

Diffusion in cellular aqueous compartmentsis determined by solute properties and thecomposition, organization, and geometry ofthe cellular compartment. We have used spotphotobleaching to measure the diffusion ofsmall solutes and a series of macromoleculesin cytoplasm, including GFP (48) and fluores-cently labeled dextrans (42) and DNAs (27).Examples of photobleaching recovery curvesare given in Figure 4a and the results are sum-marized in Figure 4b as relative diffusion ofin cytoplasm versus water (Dcyto/Dwater). Thegeneral observation is that diffusion in cy-toplasm is slowed only a few-fold comparedwith that in water, except for linear DNAs andlarge macromolecules (dextrans >500 kDa),where diffusion is greatly slowed becauseof limited movement through the actin

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Figure 4Diffusion of macromolecules in cytoplasm and mitochondria. (a) Spot photobleaching (60 × objective,short bleach time) of indicated fluorescein-labeled dextrans and linear double-stranded DNA fragmentsin cytoplasm. (b) Ratio of diffusion coefficients in cytoplasm versus saline (Dcyto/Dwater) for indicatedsolutes and macromolecules. Data taken from References 27, 42, and 48. (c) Diffusion of greenfluorescent protein (GFP) in the mitochondrial matrix. (Left) Fluorescence micrograph of matrix-targeted GFP in transfected CHO cells. (Right) Spot photobleaching (100 × lens), with Browniandiffusion model predictions for indicated GFP diffusion coefficients. Adapted from Reference 33.

cytoskeleton. An FCS study of the diffusionof dextrans in cytoplasm reported evidence foranomalous diffusion even for relatively smalldextrans (55). The DNA size-dependent re-duction in Dcyto/Dwater could be reproducedin artificial solutions containing actin net-

works (8) and was ascribed to the barrierproperties of actin networks for diffusion ofDNAs greater than their persistence length.Measured Dcyto/Dwater is reduced not only bysteric factors (molecular crowding), but alsoby the intrinsic fluid-phase viscosity of the

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aqueous environment and by binding interac-tions with mobile and fixed obstacles in cells.When these three independent factors reduc-ing diffusion of a small molecule were mea-sured independently, it was found that fluid-phase viscosity and binding accounted for lessthan 40% of total slowing, with molecularcrowding being the major determinant (23).Also, notwithstanding limitations in the ap-plication of FRAP to detect anomalous diffu-sion, fluorescence recovery curves were gen-erally monophasic and fitted well to Browniandiffusion models.

We have also found unexpectedly minorslowing of diffusion in aqueous compartmentsof intracellular organelles. The mitochondrialmatrix, a major site of metabolic processes,is the aqueous compartment enclosed by theinner mitochondrial membrane. Theoreticalconsiderations have suggested that the diffu-sion of metabolite- and enzyme-sized solutesmight be severely restricted, by more than1000-fold, in the mitochondrial matrix be-cause of its high density of proteins. However,the diffusion of GFP was only slowed aboutthreefold compared with its diffusion in water(33). Figure 4c shows mitochondrial specificGFP targeting. Spot photobleaching of GFPwith a 100 × objective (0.8-μm spot diame-ter) gave a half-time for fluorescence recov-ery of 15–19 ms with greater than 90% of theGFP mobile (Figure 4c). Predicted recoverycurves for different diffusion coefficients arealso shown; the best fitted value was 2–3 ×10−7 cm2 s−1, only ∼threefold less than thatfor GFP diffusion in water. We proposed thatclustering of matrix enzymes in membrane-associated complexes might serve to estab-lish a relatively noncrowded aqueous space inwhich solutes can freely diffuse. Subsequentmeasurement of diffusion of various GFP-tricarboxylic enzyme chimeras provided ex-perimental evidence for such a multi-enzymemacromolecular complex (17). A similar anal-ysis of GFP diffusion in the lumen of the en-doplasmic reticulum showed only ∼threefoldslowing compared with GFP diffusion in wa-ter (9). Together, these data suggest relatively

minor effects of molecular crowding on dif-fusion in cellular aqueous compartments (atleast for small solutes and relatively smallmacromolecules such as GFP). These resultsare consistent with those obtained from sim-ulations of small solutes diffusing in the pres-ence of large crowders (10).

CONCLUSIONSAND PERSPECTIVE

Molecular crowding and complexity in andaround cells can in principle produce markedslowing of diffusion as well as anomalous andcomplex diffusive behaviors, such as stronglysize-dependent diffusion. Although there areexamples of restricted and anomalous diffu-sion in cellular aqueous and membrane com-partments, greatly slowed or anomalous dif-fusion in cells is not a universal phenomenon.Experimental evidence supports the possi-bility of simple Brownian diffusion even inhighly crowded aqueous media and when thediffusing species is as large as or larger than thecrowding agent. Experimental data also sup-port the possibility of long-range nonanoma-lous diffusion of integral membrane proteins.Thus, the observation of anomalous or greatlyslowed diffusion in cells indicates the pres-ence of significant interactions, fixed obsta-cles that impede diffusion, and/or high-ordersupermolecular organization such as mem-brane rafts. Further, the possibility of facti-tious anomalous diffusion mandates consid-eration as discussed in this review with regardto experimental measurement methods andlimitations. Experimental data in cells indi-cate relatively minor consequences of molec-ular crowding at least for the diffusion offairly small solutes, such that diffusion isslowed only a few-fold (compared with dif-fusion in water) in cytoplasm and intracellu-lar organelles including mitochondria. Fur-ther refinement of the ideas presented hereis likely to follow technological advances forsingle-molecule tracking and computationaladvances for simulations of diffusion in highlycrowded media.

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SUMMARY POINTS

1. Crowding can slow the diffusion of solutes in aqueous-phase compartments and inmembranes without leading to anomalous diffusion.

2. Large reductions in solute diffusion and/or anomalous diffusion are probably indica-tors of interactions between the solute and cellular or membrane components, or offixed barriers to diffusion.

3. Crowding reduces the diffusion of small solutes and many macromolecules in cyto-plasm by only a few-fold compared to their diffusion in water.

4. Discrepancies between simulations and experiments on crowding effects on solutediffusion require further investigation.

DISCLOSURE STATEMENT

The authors are not aware of any biases that might be perceived as affecting the objectivity ofthis review.

ACKNOWLEDGMENTS

This work was supported NIH grant EB000415 and by the Research Foundation of SUNY.

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Annual Review ofBiophysics

Volume 37, 2008

Contents

FrontispieceRobert L. Baldwin � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �xiv

The Search for Folding Intermediates and the Mechanismof Protein FoldingRobert L. Baldwin � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �1

How Translocons Select Transmembrane HelicesStephen H. White and Gunnar von Heijne � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 23

Unique Rotary ATP Synthase and Its Biological DiversityChristoph von Ballmoos, Gregory M. Cook, and Peter Dimroth � � � � � � � � � � � � � � � � � � � � � � � � 43

Mediation, Modulation, and Consequencesof Membrane-Cytoskeleton InteractionsGary J. Doherty and Harvey T. McMahon � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 65

Metal Binding Affinity and Selectivity in Metalloproteins:Insights from Computational StudiesTodor Dudev and Carmay Lim � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 97

Riboswitches: Emerging Themes in RNA Structure and FunctionRebecca K. Montange and Robert T. Batey � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �117

Calorimetry and Thermodynamics in Drug DesignJonathan B. Chaires � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �135

Protein Design by Directed EvolutionChristian Jäckel, Peter Kast, and Donald Hilvert � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �153

PIP2 Is A Necessary Cofactor for Ion Channel Function:How and Why?Byung-Chang Suh and Bertil Hille � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �175

RNA Folding: Conformational Statistics, Folding Kinetics,and Ion ElectrostaticsShi-Jie Chen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �197

Intrinsically Disordered Proteins in Human Diseases: Introducingthe D2 ConceptVladimir N. Uversky, Christopher J. Oldfield, and A. Keith Dunker � � � � � � � � � � � � � � � �215

Crowding Effects on Diffusion in Solutions and CellsJames A. Dix and A.S. Verkman � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �247

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Nanobiotechnology and Cell Biology: Micro- and NanofabricatedSurfaces to Investigate Receptor-Mediated SignalingAlexis J. Torres, Min Wu, David Holowka, and Barbara Baird � � � � � � � � � � � � � � � � � � � � � �265

The Protein Folding ProblemKen A. Dill, S. Banu Ozkan, M. Scott Shell, and Thomas R. Weikl � � � � � � � � � � � � � � � � � �289

Translocation and Unwinding Mechanisms of RNAand DNA HelicasesAnna Marie Pyle � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �317

Structure of Eukaryotic RNA PolymerasesP. Cramer, K.-J. Armache, S. Baumli, S. Benkert, F. Brueckner, C. Buchen,G.E. Damsma, S. Dengl, S.R. Geiger, A.J. Jasiak, A. Jawhari, S. Jennebach,T. Kamenski, H. Kettenberger, C.-D. Kuhn, E. Lehmann, K. Leike, J.F. Sydow,and A. Vannini � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �337

Structure-Based View of Epidermal Growth Factor ReceptorRegulationKathryn M. Ferguson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �353

Macromolecular Crowding and Confinement: Biochemical,Biophysical, and Potential Physiological ConsequencesHuan-Xiang Zhou, Germán Rivas, and Allen P. Minton � � � � � � � � � � � � � � � � � � � � � � � � � � � � �375

Biophysics of Catch BondsWendy E. Thomas, Viola Vogel, and Evgeni Sokurenko � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �399

Single-Molecule Approach to Molecular Biology in Living BacterialCellsX. Sunney Xie, Paul J. Choi, Gene-Wei Li, Nam Ki Lee, and Giuseppe Lia � � � � � � � � �417

Structural Principles from Large RNAsStephen R. Holbrook � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �445

Bimolecular Fluorescence Complementation (BiFC) Analysisas a Probe of Protein Interactions in Living CellsTom K. Kerppola � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �465

Multiple Routes and Structural Heterogeneity in Protein FoldingJayant B. Udgaonkar � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �489

Index

Cumulative Index of Contributing Authors, Volumes 33–37 � � � � � � � � � � � � � � � � � � � � � � � �511

Errata

An online log of corrections to Annual Review of Biophysics articles may be found athttp://biophys.annualreviews.org/errata.shtml

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