Lateral diffusion in an archipelago

15 Pages·2005·2.69 MB·English

Lateral diffusion in an archipelago
Single-particle diffusion
Michael J. Saxton
InstituteofTheoretical Dynamics, UniversityofCalifornia, Davis, California95616; andLaboratory ofChemical Biodynamics,
Lawrence Berkeley Laboratory, UniversityofCalifornia, Berkeley, California94720 USA
ABSTRACT Several laboratories have measured lateral diffusion of single particles on the cell surface, and these measurements may
reveal an otherwise inaccessible level of submicroscopic organization of cell membranes. Pitfalls in the interpretation ofthese experi-
ments are analyzed. Random walks in unobstructed systems show structure that could be interpreted as free diffusion, obstructed
diffusion, directed motion, or trapping in finite domains. To interpret observed trajectories correctly, one must consider not only the
trajectories themselves but also the probabilities of occurrence ofvarious trajectories. Measures ofthe asymmetryof obstructed and
unobstructed random walks arecalculated, andprobabilities areevaluatedforrandomtrajectoriesthatresembleeitherdirectedmotion
ordiffusion in abounded region.
INTRODUCTION
Newtechniques ofmicroscopy have made itpossibleto ton(2,9,47),themembraneskeleton(41),ortheextra-
observe the motion ofindividual proteins orsmallclus- cellular matrix (51); any ofthese obstacles could trap
ters oflipids on the cell surface (2, 9, 11, 18, 19, 25, 28, mobile species in domains. Domain structure in mem-
43).Thesetechniquesprovideapowerfultooltocharac- branes was recently reviewed (10, 22).
terize submicroscopic domain structure in biological Little work has appeared in the biophysics literature
membranes. Inthese experiments, proteins orlipidsare onthetheory ofsingle-particle diffusion measurements,
labeled with a highly fluorescent label or with colloidal except forthe recent paper by Qian et al. (35), empha-
gold microspheres. Computer-enhanced video micros- sizing the statistical accuracy of the measurements.
copyisusedtotrackthetrajectoryofindividualparticles Someworkonthe shape ofrandom walkshasappeared
as they move on the cell surface. The time resolution is in the physics literature, often emphasizing random
typically 1/30 s and the spatial resolution is 5-50 nm. walks in high dimensions (38). Much work has ap-
Theshapeofthetrajectoriessuggestsvariousbiologically peared in the polymer literature on random walks and
important processes, such as binding to immobile spe- self-avoiding walks as models ofpolymerconformation
cies, free diffusion, hindered diffusion, directed trans- (4, 17, 44, 45). Little, ifany, ofthis work has included
port, and trapping of particles in bounded microdo- the effects ofobstructions. The work presented here is
mains. Transitions between these types of motion are inspiredinpartbytheworkofBookstein(5,6)pointing
observed. out that one-dimensional random walks are the proper
Unfortunately these trajectories can also occur, with control for time sequences ofmorphological change in
distressinglyhighprobability, inrandomwalksinanun- evolution. Thetheoryofrandomwalkswasreviewedby
obstructed system. To interpret the observed trajecto- Weiss and Rubin (50).
ries, one must consider not only the trajectories them- The paper is organized as follows. First, some exam-
selves but also the probability ofoccurrence ofvarious ples ofunobstructed random walks are given, to show
trajectoriesinobstructedandunobstructedsystems. Itis thatthereisinfactaproblem. Next,thedisplacementof
necessarytousethepropercontrol (5, 6): atwo-dimen- adiffusing particle is characterizedby histograms ofthe
sional unobstructed random walk. This problem has displacement at fixed times. Then, two parameters are
beenrecognizedintheliterature,buttherehavebeenfew used to show the asymmetry ofrandom walks. Finally,
toolsavailabletosolveit.Thegeneralproblemiscompli- methodsareproposedtodeterminewhetheranobserved
cated, so this paper will focus on two special cases, di- trajectory isaresultofdirectedmotionordiffusion,and
rected transport, and trappingin boundeddomains. whetheradiffusingparticleisinaboundedregion. Both
These special cases arebiologically significant. Identi- unobstructed and obstructed random walks are consid-
fyingdirectedmotionisessentialinstudiesofthemecha- ered. The obstacles consist of random points, cluster-
nism ofcellular locomotion (25) and the interaction of cluster aggregates, and point obstacles arranged to pro-
membrane proteins with the cytoskeleton (31). Trap- duce a bounded domain, usually hexagonal.
pingindomainsmayhaveamajorinfluenceonreaction
kinetics (23, 26, 27), and equilibrium (48). Diffusion
METHODS
mayberestrictedbygel-phaselipids(49), thecytoskele-
Diffusioncalculationsarecarriedoutasdescribedearlier(39,42). A
256X256triangularlatticeisused,withperiodicboundaryconditions.
Addresscorrespondence toDr. Michael J. Saxton, Institute of Pointobstaclesareplacedonthelatticeatrandom.Atracerisplacedat
Theoretical Dynamics, University ofCalifornia, Davis, CA a random unblocked point on the lattice, and carries out a random
95616-8618, USA. walk on unobstructed lattice sites. The calculation is repeated for
11776666 00000066--33449955//9933//0066//11776666//1155 $$22..0000 BBikopphhyyss..JJ..©uBBi.oophpyshny19sk.Society
Volume64 June1993 1766 1780
100 given configuration ofobstacles, and different random
configurations of obstacles at the prescribed area frac-
80 tion. This averaging yields the mean-square displace-
ment Kr2> as a function oft. In the usual Monte Carlo
60 calculations(12, 34, 39,40,42),thediffusioncoefficient
is obtained from the mean-square displacement, but
40 herewefocusonindividualtrajectoriesandtheprobabil-
ity oftheiroccurrence.
20 At low concentrations ofobstacles, the mean-square
displacement is
0
(r2> 4Dt. (1)
=
t The observed diffusion coefficient D is written in terms
ofthediffusioncoefficientDOintheabsenceofobstacles
FIGURE 1 Mean-squaredisplacement<r2>asafunctionoftimetfor
different types ofmotion (40). This is based in part on a figure of andadimensionless, concentration-dependentdiffusion
coefficientD*(C):
Sheetzetal.(43).TheproteindiffusioncoefficientistakentobeDo=
3x 10"-cm2/s,andthevelocityv=20nm/s,asinreference43.Fora
samplingtimeT=33ms,then,e=20nmfromEq.3c.Indimensional D = DOD*(C)9 (2)
units,therangeofthex-axisis33sandthatofthey-axis,0.040Am'.
(Curve a)Randomwalkonatriangularlatticewithnoobstructions. with D*(O) = 1. The Monte Carlo calculations are
(Curveb)Randomwalkwithanareafractionofrandompointobsta- carried out in dimensionless units
clesofC= 0.3.(Curvec)Randomwalkwithanareafractionofran-
dompointobstaclesofC=0.65,abovethepercolationthresholdof0.5 r* = rle, (3a)
for a triangular lattice. (Curve d) Directed motion, for which t* = tl-r, (3b)
(r2> = V2t2.
where e is the lattice spacing and is the jump time,
r
related by
variousstartingpositionsofthetracerwithinagivenconfiguration of
obstacles, andvariousconfigurationsofobstaclesatthesameconcen- P = 4DoT. (3c)
tration. Typically 100differentconfigurations ofobstacleswereused,
and 1000different tracers within each configuration ofobstacles. At The dimensionless mean-square displacement isthen
each time step, the position ofthe tracer is obtained, and the sums
required fortheradiusofgyrationtensorarerecorded. Atprescribed r-*2>= D*(C)t*. (4)
times,histogramsofthedisplacementandthelargestdisplacementare To simplify the notation, the asterisks are dropped un-
compiled,theradiusofgyrationtensorisdiagonalized,andhistograms
oftheasymmetryparametersarecompiled. Errorlimitsforthehisto- less it is necessary to distinguish the dimensional and
gramsarediscussedinthecaptions. Forvaluesofthediffusioncoeffi- dimensionless variables.
cient D*, independent runs gave results reproducible to 1.5% in the The usual way ofanalyzing a random walk is to plot
worstcase,and0.2-0.5%typically. the mean-square displacement as a function oftime, to
Cluster-clusteraggregatesareconstructedbystandardmethods(24, obtaincurvesasshowninFig. 1.Theseareaveragedover
30),asdescribedindetailelsewhere(42).Initially,particlesareplaced
onrandomlychosensitesataprescribedconcentration.Adjacentpar- .5000 trajectories. Unobstructed diffusion yields a
ticlesareassumedtoformclusters,andisolatedparticlesformclusters straightlineofunitslope,givingD* = 1 (curvea).Diffu-
ofunitmass.Theclustersthencarryoutarandomwalk.Whenevertwo sion in the presence oflow concentrations of random
clustersbecomeadjacent,theyaremergedintoarigidclusterirrevers- point obstacles gives, at large times, a straight line of
ilbaltyi,onwailthdipfrfoubsaiboinlictoyefofnieci.eTntheinyvemrosveleyapsraopuonrittitohnearleatfotemra,swsi.thThaetrraanns-- slope < 1, corresponding to D* < 1 (curve b). At high
concentrations ofrandom point obstacles, the system is
dom walkcontinues until only oneclusterremains, thefinalcluster-
clusteraggregate. above the percolation threshold, so that there are no
long-rangepathsfordiffusion. Thediffusingparticlesare
RESULTS
TABLE1 Diffusioncoefficients asafunctionofobstacle
Trajectories of random walks
concentration
WeconsidertrajectoriesobtainedfromMonteCarlocal-
C D*
culations. In these calculations, lattice sites are blocked
atrandom, andtheobstacleconcentrationisgivenasan Obstacles 0.0 0.999
area fraction C, equaltothe fraction ofblocked sites. A 0.1 0.819
point tracer carries out a random walk on unblocked 0.2 0.629
0.3 0.425
sites ofthelattice, anditsdisplacement from itsstarting
0.4 0.210
positionisrecordedatprescribedtimes, yieldingthedis-
placement r2 asa function oftime t. Displacements are Cluster-clusteraggregates 0.1 0.316
0.3 0.0840
averaged overdifferent random starting points within a
SSaaxxttoonn LLaatteerra.lDDiiffffuussiioonnooffSSiaingglleePPaarrttiicclleess 11776677
. . * l*F * * . lF lF
a =0.2426
A2 =0.3716
I * I. .1I a a I I I I I I I I
I . . I" . A. I . .
11776688 BBiioopphhyyssiiccaallJJoouurrnnaall VVoolluummee6644 JJuunnee11999933
FIGURE2 Displacementr'asafunctionoftimet(indimensionlessunits)forasequenceof10randomwalksonanunobstructedtriangularlattice.
Tosavespace,scalesarenotshown.Thex-axiscorrespondsto 1024timesteps;majordivisionsare256units.They-axiscorrespondstor2=3072;
majordivisionsare 1024units.Thestraightlineis<r2>=t,themean-squaredisplacementindimensionlessunitsaveragedoveralargenumberof
randomwalks.Foreachrandomwalk,amapofthedistinctsitesvisitedbythetracerisshown,alongwiththecenterofmassoftherandomwalk
(filledcircle),theellipseofgyration, andtheasymmetryparametersa2andA2,alldiscussedlater.
trapped in bounded domains, so that Kr2> levels offat ance as the random walk is magnified, until individual
large values of t, and the limiting value r2(oo)> is a stepsonthelatticecanbeseen. Self-similaritywasnoted
measureofthesizeofthedomains(curvec). Indirected by Perrin in measurements of Brownian motion pub-
transport,theobservedspeciesisattachedtomobilecyto- lished in 1909 (see reference 29). Butifthere are obsta-
skeletalelementsoriscarriedalonginabulkmembrane cles with structure on some length scale, the random
flow.Thetracermovesinanondiffusivemanneraccord- walk will show structure on the corresponding time
ingto r = vt, where v is velocity (curve d). scale.
Data analysis requires a value of D. To interpret
Monte Carlo data, dimensionless variables are used Distribution of displacements
(Eqs. 1-4). Table 1 gives values ofD* obtained from One method ofanalyzing single-particle diffusion mea-
Monte Carlo calculations andEq. 1 forvariousconcen- surements uses histograms of the displacement at
trations ofobstacles. Tointerpretexperimental data, di- various fixed times (2). The solution of the diffusion
mensional variablesareused. ThevalueofDusedought equation in two dimensions for an instantaneous point
tobeanaverage value, obtainedfrom fluorescence pho- sourceattheoriginatatimet= 0is(reference8,p. 258)
tobleachingrecoverymeasurements, orfromsingle-par-
tbiecrleodfitfrfaujseicotnormieesa.surementsaveragedoveralargenum- C(r, t) = exp(-r2/4Dt), (5)
47rDt
rieFs;igF.ig1.s2hoswhsoawvseraagseesquoevnecreaolfar1ge0niundmibveidruaolftrraajnedctoom- where C(r, t)istheconcentration andDisthediffusion
coefficient. The probability density is
walks. In Fig. 2, r2 is given as a function oftime t; the
irregularblobsare mapsofthe sitesvisitedbythetracer. P(r, t)dr= 2irrdrC(r, t), (6)
Notetheapparent structure. Fig. 2 iandthe firstpartof
Fig. 2 a suggest free diffusion, at much different rates; andthe mean-square displacement is
Fig. 2, b, c, andgsuggeststrappinginaboundedregion;
Fig. 2 dsuggests diffusion in abounded region followed Kr2(t)> = 2wrdrC(r, t)r2 = 4Dt, (7)
byaperiodofrapid motion, perhapsdirectedtransport.
Yetallthesearepurelyrandomwalks, withnoobstacles so that
whatsoever, and no mechanism for directed motion.
Note the frequency of random walks with apparent
structure. Thisisnotapathological setofrandomwalks P(r, t)dr Kr(t exp[r/<r(t)>I. (8)
partice
selectedto showextremes, butatypicalsequencegener- The p obablity hat sdif dadit
ated by the Monte Carlo program. The probability that a particle has diffused a distance ro
A random walk has no characteristic time scale, and orgreater at atime tis
the fluctuations seen in Fig. 2 appear on all time scales
(16). Fig. 3 shows a single random walk on an unob- Pr(r2 ro) = P(r, t) dr = exp[-r2/<r2(t)>]. (9)
structed lattice at three different time scales. The ran-
dom walk is self-similar; there is no change in appear- This equation provides onetest fordirected motion.
SSaaxxttoonn LLaatteerra.lDDiiffffuussiioonnooffSSiinngglleePPaarrttiicclleess 1769
16 ofD*, Eq. 8 describes the Monte Carlo results well. At
moderate concentrations, random point obstacles de-
crease the rate ofdiffusion but do not change the shape
12
ofthe probability distribution. But ifthe tracers are ob-
c'J structed by a cluster-cluster aggregate at the same area
8 fraction, Eq. 8 does not fit the data well, because the
0T4U- positions of the obstacles are correlated, not random
(42). Fig. 4 c shows Monte Carlo results for the aggre-
4 gate, and curves from Eq. 8 with D* from Table 1.
When long-range diffusion isblocked, the behaviorof
thecurveschanges. AtC= 0.5,thepercolationthreshold
0
4 for the triangular lattice, Eq. 8 is not applicable. Monte
t/1024 Carloresultsareshown inFig. 4d. Atthisconcentration
ofobstacles, sometracersareonpercolatingclustersand
4 havelong-rangediffusion pathsopentothem;theproba-
bility distribution for these tracers follows a generaliza-
tion of Eq. 8 (21). Other tracers are trapped in finite
3
clusters, and yield a different probability distribution.
TheobservedP(r, t)isan averageofthesedistributions.
CM
2 ThepersistentpeakatlowRisduetotrappedtracers. At
CTt-
C = 0.6, allthe tracersaretrappedin finite clusters, and
\L
the probability density reaches a limiting value with
time, as shown in Fig. 4 e. Similarly, fortracers trapped
inahexagonofradius 15,theprobabilitydensityreaches
0 alimitingvalue, asshowninFig.4f, buttheshapeofthe
0.0 0.25 0.50 0.75 limiting distribution is different, reflecting the different
t/ 1024 sizes and shapes ofthe domains.
1.0
c Asymmetry of random walks
0.75 The average behavior ofa diffusing particle is symmet-
ric: a Gaussian distribution growing shorter and wider
cxJ
withtime(Eq. 5). Buttherandomwalksmakingupthat
CTt- 0.50 _ average tend to be highly asymmetric, simply because
CL there are many more asymmetric conformations than
0.25 symmetric ones (37). In this section we consider two
parameters that characterize the asymmetry ofrandom
walks, andusethemto showthatasymmetryistherule,
0.00L. Ant_~~64~~~- - 128 192 2516 not the exception. These measures of asymmetry are
useful in recognizing directed motion, because directed
motion produces averyelongatedtrajectory. The asym-
FIGURE3 Displacementr2asafun(ctionoftimeforonerandomwalk metry of random walks was reviewed by Rudnick and
(in dimensionless units) on an unolbstructed lattice at three different Gaspari (38).
timescales.Theboxesinaandbshc)wtheportionoftherandomwalk For a random walk of n steps, the two-dimensional
enlargedinthenextgraph. Bothscaleschangebyafactorof4ineach radius ofgyration tensoris (38, 45)
enlargement.
(x<> >x>V xy)>-x (10)
In Fig. 4, Monte Carlo results are compared with Eq.
8. Consider the results when long-range diffusion is al- where the averages are over all n steps in the random
lowed. Fig. 4 a shows P(r, t)dr for unobstructed diffu- walk: Kx> = (1/n) Iin xi, andsoforth. Thistensorcan
sion from Eq. 8, andtwosetsofdatapointsfromMonte be diagonalized by a rotation through an angle
Carlo calculations. Agreement is good, with deviations
resulting from statistical noise, binning, andthediscrete 2 ( r- T7 ) (1
structure ofthelattice. Fig. 4 bshowssimilarcurvesand
Monte Carlo data forrandom point obstacles at an area The principal radii ofgyration aretheeigenvalues ofthe
fraction C= 0.3, with D* from Table 1. Withthisvalue tensor T,
11777700 BBiioopphhyyssiiccaallJJoouurrnnaall VVoolluummee6644 JJuunnee11999933
R2, R2='/2[(T,x+ Tyy) ± X(T-TB)2 + 4Ty], (12) andthemostprobable valueisa2 = 0.12, corresponding
toR2/RI = 0.35. Randomwalksarelesssymmetricthan
and the radius ofgyration is percolation clusters, forwhich <a2> 0.4 (15).
=
R2 = R2 + R2.(13) Another measure ofasymmetry isthe parameter (36,
38)
One way to illustrate the asymmetry ofthe trajectory is
tohreieenltleidpseatofagnyraantgiloen,X,anaenldlipcseentoefrseedmioanxetsheR,ceanntderR2o,f 2 - ((RR22-+RR22))22 ' (15)
mass ofthetrajectory. Examples are given in Fig. 2.
One measure ofasymmetry isthe ratio ofthe smaller The numerator measuresthedeviation from circularity;
to the largerprincipal radius ofgyration (15 the denominator normalizesthedeviation bythe radius
), ofgyration. This parameter is equal to 0 for circularly
a2 = R2IR2 (14) symmetrictrajectoriesand forlineartrajectories. Rud-
1
jLeicnteoariretsrahjaevcteorai2es=ha1v.eNao2t=e0t;hactirtcuhlearalvyersaygmemsetirniTctarrae- fnRioc2rk)m2a>fno1rd(hGRiags1h2p+adirRmi2e)(n23s>8i,)onfesom,rpwbhhuaitscithzheetyhoecnyoeonbsativadeiernragaeKn,A2a<n>(atRloyItbi-ec
tsiuesmsofovaetrratjheecptoorsyi,tinoontatofeatchhetdiismteipnoctinsti,teasnvdisairteedp.roCpoenr-- the better parameter to characterize a random walk.
Histograms ofA2 show little time dependence when
siderthreeunblockedpointsforminganequilateraltrian- the obstacle concentration is below the percolation
gle, Aat(0, 0), Bat(1, 0), andCat(1/2, i/2), withall
threshold. Below the threshold, the histogram is again
adjacent points blocked. Two different trajectories with
the same points visited and the same endpoints, say Fiingd.e6p.en(dAeganitn,ofctohneceonbtsrtaatciloencodnecpeenntdreatnicoen,isaasppsahroewnntiant
ABCABC and AAAABC, have different values of<x>,
<Y>,* * *andtherefore different asymmetryparameters, C = 0.45 but not at C = 0.40.) As Cincreases, the frac-
tion ofpoints with A2 increases; these result from
rheecrTteheeda2me=iogte1inoavnnaldfuear2orma=tdi1io/f2af,u2rseiissopnue,scetbfiuuvtleliynn.otdiisntirnegsuoilsvhiinnggtdhie- tArlatcheorsugthraap2paedndoAn2iasroel=attrei1dviaploliyntrselaatnedd fpoarirasnyofinpdoiinvtisd.-
structure ofdifferenttypesofobstacles. Fig. showshis- ual trajectory, their averages over a set of trajectories
5 are not.
tograms ofa2. For unobstructed diffusion anddiffusion
with random point obstacles below the percolation The histograms ofA2 are in excellent agreement with
dthernetshvoalldue,,tahsescuhrovwenrianpiFdilgy.c5oan.vBerogtehsRt2oaantidmRe2-iinndcerpeeans-e tnhaomsiecsobctaalicnueladtifoornsafroraandbaolmlapnodlysmperirnbgymoBdreolwn(i4)a.nTdhye-
withtime,buttheirratioquicklybecomestimeindepen- curve is constant for small A2> 0, and falls offas A2
pdeenntd.enBteloofwotbhsetapcelrecocloantcieonntrtahtrieosnh,olads,sthheowcnurivneFiisgi.nd5eb- ttaiopmpefrsoo,ramc<hAee2xs>te1,ndi0em.dp3l9s7yt.irnucgtutrheast.rFaonrdoCm=wa0l.k0s-0a.r4e,uantlilkaerlgye
afoprprCoa=ch0e.s0-t0h.e4.peBructo,laatsiotnhetchroenscheonltdr,attihoensohfapoebsotafctlhees In Fig. 2, a=symmetry parameters and ellipses ofgyra-
curvechanges,asshowninFig. 5 bforC= 0.5.(Changes tpiroonpearrteiesshoofwtnhefotrratjheecttorrayj,ecntootrioesf.thTehesesteoqfusaintteistviiessitaerde,
0in.4s5habpuet anroetafpopraCren=t0a.t40t.h)eFpoeradkifoffutsihoenciunrvaeffinoirteCre=- wsooutlhadtydiieflfderdeinftfetrreanjtecvtaolruieess oofvetrhetsheeqsuaanmteitiseets.ofpoints
gion, the curve is strongly time dependent, as shown in
pFirgo.ba5bicliftoryadihsetrxiabguotniaolnrdeosmeamiblnesofthreadciuursv7e.fIonritainalulnyotbh-e What is the probability of a fast
trajectory?
structed random walk, but as time increases the curve
shiftstoaformreflectingtheshapeofthedomain. Fig. 5 To determine whether an observed trajectory is the re-
dcompares the histograms fordiffusion in the presence sult ofdirected transport or random motion, one can
ofrandom point obstacles andcluster-clusteraggregates evaluatea2orA2,andcomparethemeasuredvalueswith
atanareafraction of0.3. Theaggregatesaremuchmore the calculated probabilities. It is more convenient to do
effective obstacles than random pointsare(42); thedif- this using the cumulative probability curves of Fig. 7,
fusion coefficients are D* = 0.425 for random points calculated from the data in Figs. 5 and 6. These curves
and D* = 0.0840 for aggregates. Despite this difference give the fraction oftrajectories with an asymmetry pa-
inD*,thehistogramsaresimilar,indicatingthata2isnot rameter at or below a given value. For random point
a sensitive probe ofobstacle structure. obstacleswith C 0.4, thecumulative curvesshowvery
Whatthisparametershowsisthatasymmetry iscom- little dependence on concentration (Fig. 7 a) or time
mon.Therearefewlinearorcircularlysymmetrictrajec- (Fig. 7 b). The curves are smooth; most ofthe noise is
tories. For diffusion in the presence of random point averaged out. Recall that an extended trajectory corre-
obstacles below the percolation threshold, the average spondsto a2 andA2 1. Numerical values ofthe
value is <a2> = 0.28, corresponding to R2/R, = 0.53, cumulative probability for small a2 and large A2 are
SSaaxxttoonn LLaatteerraailDDiiffffuussiioonnooffSSiinngglleePPaarrttiicclleess 1771
0.1I.I1as5 I I - I' 00..3
a
0.10k 0.2
t=128
'a
;L
a-
0.05 0.1
=4096
on
0.0 D 20 40 60 80 100 0 10 20 30 40 50
r r
t=128
L0 L0
0O _0
0.1
0.0 0.0
0 20 40 60 80 100
0.3 I I I
f
0.2
-
L-
L- t=128
4-
0.1
0.0 v .... ........-- I 0.0 1I I . I I
0 20 40 60 80 100 I 10 20 30 40 50
0
r
FIGURE4 ProbabilitydensityP(r, t)drfrom Eq. 8. Notethechangeofscalebetweenthefirstthreeandthelastthreegraphs. Histogramswere
calculatedforAr=0.01.(a)Unobstructeddiffusion,C=0.0.Lines,fromEq.8fortimest= 128,256,. . ,4096.Crosses,MonteCarloresultsfor
t= 512and4096.(b)DiffusioninpresenceofrandompointobstaclesatanareafractionC=0.3.Lines,fromEq.8fortimest= 128,256,...
4096.Crosses, MonteCarloresultsfort= 512and4096.MonteCarloresultsfortwoindependentrunsforC=0.0andtwoindependentrunsfor
C=0.3werealmostindistinguishableonthescaleofthefigure.(c)Crosses,MonteCarloresultsforacluster-clusteraggregatefort=512and4096
atanareafraction C= 0.3. Lines, from Eq. 8. (d)MonteCarloresultsforrandompointobstaclesatanareafraction C= 0.5,thepercolation
thresholdforsitepercolationonthetriangularlattice,fort= 128,256,and 16,384.(e)MonteCarloresultsforrandompointobstaclesatanarea
fractionC=0.6,abovethepercolationthreshold,fortimest= 128,256,and 16,384.(f)MonteCarloresultsforaboundedregion,ahexagonof
radius 15,fortimest= 128,256, 512,and 16,384.
given in Table 2 (from two independent runs for C = length. Thisis alimitation ofthe lattice model. Instead,
0.0, t = 16,384). considerazigzagtrajectory, asshown intheinsetofFig.
Purely linear trajectories are not a useful example of 7 c. Itissimpletocalculatea2foraseriesofthesetrajec-
directed motionbecausea2 = 0andA2 = 1,regardlessof tories; the fraction ofrandom-walk trajectories at least
1772 BBiioopphhyyssiicca.lJJoouumma.l VVoolluummee6644 JJuunnee11999933
a
0 Co
0o
LL
0.011
a2 a2
c
00 c0
LIL LL
0.0 I"_-d--- I-I-
0.0 0.2 0.4 0.6 0.8 1.0 1.0
a2 a2
FIGURE5 Histogramsoftheasymmetryparametera2forrandomwalksinthepresenceofvariousobstacles. Inthehistograms, Aa2 =0.01.(a)
Time dependence forunobstructed diffusion. C= 0.0, t = 128, 256,..., 16,384. Histograms for C< 0.4 showsimilartimedependence. The
thicknessofthecurveisareasonableestimateofstatisticalerror.CurvesforoneindependentrunforC=0.0andtwoindependentrunsforC=0.3
wereindistinguishablefroma.(b)Concentrationdependenceatfixedtime.C=0.0,0.1,..,0.5;t= 16,384.(c)Timedependenceofhistogramfor
abounded region, ahexagon ofradius7, fort = 16, 64,..., 16,384. (d)Comparison ofrandom pointobstacles(solidline)andcluster-cluster
aggregates(dashedline)atanobstacleconcentration C= 0.3andt= 16,384.
thatelongatedcanthenbefoundfromFig. 7a. Valuesof trappedinafinitedomain. Todeterminewhethersucha
a2 and Pr(a2) (interpolated from Table 2) are shownin trajectoryreallyindicatestrapping,weneedtheprobabil-
Fig. 7 c, along with the probabilities from Eq. 9. Both ity I(R, t) that an unobstructed random walk remains
probabilitiesdecreaseveryrapidlywithlength.Forexam- withinaregionofradiusR foralltimes < t. Thisproba-
ple, a zigzag trajectory ofX-displacement 12 has a2 = bility is derived in Appendix A.
0.0144, andtheprobabilitythatatrajectorysoelongated Fig. 8 a shows probability distributions from Eq. A3
will occurbychanceinanunobstructedrandomwalkis andMonteCarloresultsforunobstructeddiffusion. Fig.
0.0025 from Pr(r 2 ro) and 0.0015 from Pr(a2). 8 b shows similarresults fordiffusion in the presence of
In testing whether an observed trajectory is likely to random point obstacles at an area fraction of0.3. The
occur in a random walk, assume that the trajectory is smalldifferencebetweentheMonteCarloresultsandthe
random.Anunobstructedrandomwalkhasnomemory, theoretical curves is presumably the result of lattice
sothe origin oftimecanbechosen atwill. Take t = 0to structure andbinningintohistograms. Whentheappro-
bethe startofthetrajectory segment in question, calcu- priate diffusion coefficient D* is used, Eq. A3 can be
latea2 forthatsegment, andthendetermine from Fig. 7 used for C < 0.3, but as Capproaches the percolation
how probable that shape of segment is in an unob- threshold, Eq. A3 no longerapplies.
structed random walk. InFig. 8c, foraparticletrappedinahexagonofradius
What is the probability that a random 7, the cumulative probability reaches a time-indepen-
walkwill staywithin a bounded region? dreesnutltsvafloureabycltus=ter1-c0l2u4s.teFriga.gg8redgastehoawtsCMo=nt0e.3,Caarnldo
Fig. 2 shows several random walks in an unobstructed curves from Eq. A3 for random point obstacles at the
system where the tracer remains in a small region for a same concentration. The two sets ofcurves are similar
long time, giving the appearance that the tracer is initially, but around t = 256, the curve for aggregates
SSaaxxttoonn LLaatteerraallDDiiffffuussiioonnooffSSiinngglleePPaarrtfiicclleess 11777733
0.020
0.015
c
t0s
a
- 0.010 cx
-
0.005
C)
0.0 -. 0.2 0.4 0.6 0.8 1.0
A2 Asymmetryparameter
0
FIGURE 6 Histograms ofthe asymmetry parameterA2 for random
walksinthepresenceofvariousconcentrationsofrandompointobsta-
cles, forC= 0.0, 0.1, . . ,0.5; t = 16,384. Inthehistograms, AA2 =
0.01. The peak atA2 = 1 is from tracers trapped on isolated points.
Again,thethicknessofthecurveisareasonableerrorestimate.
Cu
startstolagthecurveforrandompointobstaclessignifi-
cantly. It was observed earlier that the difference be-
tweenthesetwotypesofobstaclesisshown mostclearly
in long-range diffusion measurements (42).
Suppose that a particle is observed to remain in a re-
gion R C 10. Assume C = 0.0, so D* = 1, and Fig. 8 a
applies. If the particle has been observed for 64 time CW Asymmetryparameter
steps, theprobability thatithasremained within the re-
gion R 10 is 0.63, and one cannot tell whether the
0
particle istrappedornot. Ifitremainstherefort = 256,
the probability is0.04, suggestingtrapping. Ifitremains
therefort = 512, theprobabilityis0.0010, andtrapping
islikely. Ifrandompointobstaclesarepresentatanarea
fraction C= 0.3,Fig. 8bapplies,andasimilarargument
holds, but one must observe the particle longer to con- 0~
clude that it is trapped. If the particle is trapped in a
hexagonal region of radius 7, Fig. 8 c shows that the
probability is 0.25, even atlongtimes.
Thisanalysiscanbesimplifiedbyplotting I(R,t)asa
function oftatfixed valuesofR, togivetheprobability
thatadiffusing particle staysinaregionR asafunction
Xdisplacement
oftime,asshowninFig. 9a. But,accordingtoEq.A3,
isa function ofDt/R2 alone (or, in dimensionless vari-
FIGURE 7 Cumulative probability curvesforasymmetry parameters
ables, D*t*/4R*2). Sothisfamilyofcurvescanbeplot- a2andA2. (a)Concentration dependence. Curvesareshown forran-
tedasafunctionofDt/R2andreducedtoasinglecurve, dompointobstaclesatconcentrations C= 0.0,0.1,0.2,0.3, and0.4,
as shown in Fig. 9 b. For Dt/R2 > 0.1, this curve is a for a time t = 16,384. (b) Time dependence. Curves are shown for
straight line: unobstructeddiffusion,C=0.0,fort= 16andt= 16,384;intermediate
timesgive intermediate curves. (c) Example. Logarithm ofa2 (solid
log = 0.2048 - 2.5117(Dt/R2). (16) l(idnaes)haenddlitnhee)pfrroobmabEilqi.ti9esfoPrrz(iag2z)ag(dtortajteecdtolriinees)offrvoamribouasndlenPgrt(hrs.2Trhoe)
insetshowsazigzagtrajectorywithanXdisplacementof4.Theparticle
So, in the example ofthe particle in the region R* < isassumedtomovefromlefttoright, movingonelattice constantat
10, ifD* = 1 and t* = 64, D*t*/4R*2 = 0.16, and the eachtimestep,withoutretracinganyofitspath.Foreachtrajectory,a2
probability = 0.63canbeobtainedfromFig. 9b. Ifwe iscalculatedfromEqs. 10-14,andtheprobabilityofthatvalueofa2is
assume, asinthecaptiontoFig. 1, that e = 20 nm, so = found from the cumulative probability curve in b for C = 0.0, t=
1/30 s, andD = 3 X 10-1" cm2/s, thentheradiusofthe 16,384.
11777744 BBiioopphhyyssiiccaallJJoouurmnaall VVoolluummee6644 JJuunnee11999933
TABLE2 Cumulativeprobabilitiesofasymmetryparameters obtained from the solution to the diffusion equation, as
shown in Appendix B. In dimensionless form, this is
a2 Pr(a2) A2 Pr(A2)
0.01 0.00027 0.80 0.95865 r2>/Kr2(o)> = 1 + F(t/ro), (19)
0.02 0.00320 0.81 0.96497 where Kr2(oo)> is given by Eq. 17, To = Kr2(oo)>/D,
0.03 0.01079 0.82 0.97067 and the function F is defined by Eq. B14. Values of
00..0054 00..0024422886 00..8834 00..9987052673 r2(0o )>for some simple domain shapes are given in
0.06 0.06462 0.85 0.98438 Eqs. B18.
0.07 0.08924 0.86 0.98794 This equation shows that the Monte Carlo results for
0.08 0.11607 0.87 0.99080 unobstructed diffusion in circular regions can be re-
0.09 0.14382 0.88 0.99316
duced to a common curve by replotting the results in
0.10 0.17293 0.89 0.99523
0.11 0.20167 0.90 0.99682 terms of<r2>/Kr2(oo)>and t/To, asshownin alog-log
0.12 0.23037 0.91 0.99809 plotin Fig. 10 c. Asexpected, thepoints fordiffusion in
0.13 0.25833 0.92 0.99889 hexagonal regions followEq. 18 closely, asdothepoints
0.14 0.28612 0.93 0.99952 for elongated hexagons of moderate elongation. For
0.15 0.31323 0.94 0.99981
0.16 0.33963 0.95 0.99995 these regions D* = 1, and Kr2(Co)) was obtained from
0.17 0.36543 0.96 0.99999 the appropriate radius ofgyration ofthe accessible re-
0.18 0.39088 0.97 1.00000 gion ofthe lattice. For diffusion in a hexagonal region
0.19 0.41514 0.98 1.00000 containing random point obstacles, Eq. 18 is followed
0.20 0.43871 0.99 1.00000 for C < 0.2, anddeviations appear for C = 0.3 and 0.4.
HereD* < 1 wasthevalueforalarge system ofrandom
point obstacles at the prescribed concentration (Ta-
region is 200 nm, t* = 64 corresponds to 2.1 s, andDt/ ble 1).
R2 = 0.16. The results forrandom point obstaclesabovethe per-
colation threshold do not fall on the common curve.
How long does a particle require to Thissimple scalingtreatmentdoesnotapplytorandom
explore a bounded domain? point obstacles above the percolation threshold, where
Whenadiffusingparticleistrappedinafiniteregion,the thereisadistributionofirregularlyshapeddomainswith
mean-squaredisplacementincreaseswithtimetoalimit- irregularboundaries. Thecurvesfor r2(t)>areofadif-
ing value (Fig. IO a) ferent form (7, 20), andthiscaseisbeyondthescope of
thispaper.
Kr2(oo)> = 2R , (17) Forparticlestrappedin compactregionswith smooth
where R2 is the radius ofgyration ofthe region (32). boundaries, then, Eq. 19 can be used to estimate the
Thisrelationholdsforbothlatticeandcontinuumdiffu- observation time required foratracerto seethebound-
aries of a domain ofgiven size. The break point is at
sion.
Itismoreinformativetoshowthetimedependenceof t/To = 1, and the curve levels offat t/r0 = 4. At moder-
tb)h.e mTehaen-insiqtiuaalrceudrivsepdlraecgeimoenntthaesnaylioegl-dlsoagsptlroatig(hFtigl.in1e0, faitneitceodncoemnatirnatrieoancsheosfiotbssltiacmlietsi,ngaddiisfpfluasciengmepnatrtaitclaetiinmea
and the curved transitional region is small. The initial t - 4<r2(00)>ID. (20)
part ofthe curve isgiven bythegeneralization ofEq. 1,
An estimate oft = <r2>14D based on Eq. 1 is signifi-
(r2> oC t2/dw, (18) cantly low, even for compact regions. Ifthe domain is
very elongated, the characteristic time isgreater (7).
where dwistheanomalousdiffusion exponent (20). For
unobstructeddiffusion inaboundedregion, d, = 2, and
DISCUSSION
theinitialpartofthecurveisofslope 1,justasforunob-
structed diffusion (Eq. 1). For random point obstacles The main point ofthis paper is that to interpret single-
above the percolation threshold, however, the initial particlediffusionmeasurementscorrectly,onemustcon-
slope is <1, and dw > 2. This isanomalous diffusion, in siderthe probability that an observed trajectory will oc-
which diffusion is slowed by the presence of"dangling cur, notjust the trajectory itself. One must look at the
ends, bottlenecks, and backbends" (3). Nagle (33) dis- statisticsofrandomwalks,justasonemustuseastatisti-
cussestheuseofalogarithmictimescaletoexaminethe cal treatment to interpret single-channel conductance
effectoflong-timetailsin measurementsoflateraldiffu- measurements.
sion by fluorescence photobleaching recovery. A comparison ofanalytical and Monte Carlo results
Ifadiffusing particle istrapped in acircularregion of shows that for unobstructed diffusion, the analytical re-
radius a, the mean-square displacement Kr2(t)> can be sultsapply, withD* = 1,andforrandompointobstacles
SSaaxxttoonn LLaatteerraallDDiiffffuussiioonnooffSSiinngglleePPaarrttiicclleess 1775