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Annu. Rev. Biophys. Biomol. Struct. 1997. 26:37399

Copyright c 1997 by Annual Reviews Inc. All rights reserved

SINGLE-PARTICLE TRACKING:Applications to Membrane Dynamics

Michael J. SaxtonInstitute of Theoretical Dynamics, University of California, Davis, California 95616;

email: [email protected]

Ken Jacobson

Department of Cell Biology and Anatomy, University of North Carolina at ChapelHill, Chapel Hill, North Carolina 27599; email: [email protected]

KEY WORDS: single-particle tracking, fluorescence recovery after photobleaching, lateraldiffusion, membrane dynamics, cell membrane

ABSTRACT

Measurements of trajectories of individual proteins or lipids in the plasma mem-

brane of cells show a variety of types of motion. Brownian motion is ob-

served, but many of the particles undergo non-Brownian motion, including di-

rected motion, confined motion, and anomalous diffusion. The variety of motionleads to significant effects on the kinetics of reactions among membrane-bound

species and requires a revision of existing views of membrane structure and

dynamics.

CONTENTS

PERSPECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

Capabilities of SPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

Modes of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

EXPERIMENTAL TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

DATA ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Classification of Modes of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380Anomalous and Normal Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381Confined Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385Directed Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389SPT and FRAP: Effects of the Label . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

WHAT DIFFUSION TELLS US ABOUT MEMBRANE STRUCTURE . . . . . . . . . . . . . . . . 391

TECHNICAL PRIORITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

373

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374 SAXTON & JACOBSON

PERSPECTIVES

In single-particle tracking (SPT), computer-enhanced video microscopy is used

to track the motion of proteins or lipids on the cell surface. Individual molecules

or small clusters are observed, with a typical spatial resolution of tens of

nanometers and a typical time resolution of tens of milliseconds. Some generalquestions addressed by the technique are as follows:

(a) How do particles move on the cell surface? To what extent does the mo-

tion of various particles deviate from pure diffusion? How is that motion

controlled, and what is its function?

(b) How is the cell surface organized? To what extent do membranes deviate

from the fluid mosaic model? Is a fractal time model a useful description of

the cell surface (36, 63)? How are structures on the cell surface assembled?

Does compartmentation prevent crosstalk of receptors (30)? What regional

or global control over cell membrane dynamics exists (85)?

(c) What are the effects of heterogeneous motion in a heterogeneous environ-

ment on kinetics and equilibrium (3, 20, 44, 91, 99)?

More specifically, SPT may help to answer questions about particle motion

raised by fluorescence recovery after photobleaching (FRAP) measurements.First, FRAP experiments show that diffusion coefficients for proteins in a cell

membrane are 5100 times lower than the values for proteins in an artificial

bilayer (28, 103). Many mechanisms may be involved: obstruction by mobile

or immobile proteins, transient binding to immobile or mobile species, confine-

ment by membrane skeletal corrals, binding or obstruction by the extracellular

matrix, and hydrodynamic interactions. These mechanisms have been difficult

to sort out, in large part because some or all of them may occur simultaneously,

and their relative importance may depend on the protein and the cell type (30).

Second, a significant fraction of protein and lipid is immobile on the time scaleof a FRAP experiment. For artificial bilayers and rhodopsin in the rod outer

segment, recovery is close to 100%, but in the plasma membrane, recovery is

typically 25% to 80% (30). The increased resolution of SPT ought to make it

possible to understand the FRAP immobile fraction. Third, in FRAP exper-

iments, the distribution of observed diffusion coefficients D is much broader

than expected from experimental error (28, 51, 98). Values of D vary around

twofold among different points on a single cell, and tenfold among cells (51).

This suggests significant heterogeneity in the membrane, a view supported by

other evidence (7, 28, 29).

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SPT IN MEMBRANES 375

Capabilities of SPT

SPT has several advantages over FRAP measurements. The spatial resolu-

tion is approximately two orders of magnitude higher than FRAP, so that with

sufficient time resolution (65) motion in small domains can be characterized.

Typically the time resolution is similar to FRAP, so the minimum detectablediffusion coefficient is lowered by approximately two orders of magnitude. Fur-

thermore, FRAP averages over hundreds or thousands of diffusing molecules,

but SPT measures individual trajectories. Thus, different subpopulations indis-

tinguishable by FRAP can be resolved. SPT provides the ultimate specificity

in measurement of motion of membrane components, particularly if the in-

dividual particle tracked could be characterized in terms of, for example, its

phosphorylation state.

Modes of MotionA major advantage of SPT is the ability to resolve modes of motion of in-

dividual molecules, and a major result of the technique is that motion in the

membrane is not limited to pure diffusion. Several modes of motion have

been observed: immobile, directed, confined, tethered, normal diffusion, and

anomalous diffusion. In an ensemble average, the time dependence of the

mean-square displacement (MSD) for pure modes of motion is much different

(Figure 1) so the motion can be classified readily.

Figure 1 The mean-square displacement r2 as a function of time t for simultaneous diffusion

and flow, pure diffusion, diffusion in the presence of obstacles, and confined motion.

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Two important phenomena have been observed that are related to the classi-

fication of modes of motion. First, correlated motion can provide convincing

evidence that apparently non-Brownian motion is in fact non-Brownian mo-

tion and not merely a fluctuation in Brownian motion. Second, practically all

experimental results show apparent transitions among modes of motion. If atransition is real, it could result from partition of the mobile species into dif-

ferent microdomains or from an active control mechanism such as transient

binding to a cytoskeletal motor (76, 90).

History

In the first SPT experiment on cell membranes, Barak & Webb (5) tracked

a fluorescent-labeled low density lipoprotein receptor (see also 42, 45). De

Brabander et al developed the technique of nanovid microscopy, in which

a highly scattering colloidal gold label is used with bright-field microscopy(24). They applied the technique to endocytosis and protein motion on the cell

surface (25). Sheetz and collaborators developed techniques using differen-

tial interference contrast microscopy to determine particle coordinates with

nanometer resolution, and applied this to the motion of motor molecules and

membrane proteins (41, 81, 89). This combination of techniques led to current

SPT work on gold-labeled membranes.

EXPERIMENTAL TECHNIQUES

Video microscopy is reviewed in (48, 49, 92), and SPT techniques, resolution,

and error analysis are discussed in several reviews (6, 4143, 65, 78, 81, 88,

89).

Nanometer-scale SPT is possible because the center of a small particle can

be located with a precision well below the wavelength of light, even though

two particles at that separation cannot be resolved (43, 81, 88). The particle

is much smaller than the wavelength of light, so its image is an Airy disk, and

two nearby particles give partially overlapping Airy disks. According to the

Rayleigh criterion (49), if the particles are too close, the pair cannot be resolved.But this unresolved spot is more intense than the spot for a single particle, so

the number of particles can be determined, at least well enough to distinguish

multiple particles from a single particle. For a wavelength of 546 nm and a

numerical aperture of 1.4, the radius of the Airy disk is 238 nm.

The limiting spatial accuracy in an SPT measurement is set by the mechanical

stability of the apparatus and is obtained from trajectories of stationary parti-

cles. The scatter in position is 130 nm, yielding a minimum observable D of

5 1014 to 5 1013 cm2s1. For mobile particles, the spatial accuracy is

decreased by the motion of the particle during the acquisition time of the image,and it is therefore a function of D (81). The acquisition time depends on the

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label. For gold labels, images are usually obtained at the standard video rate,

so the image is integrated over 1/30 or 1/25 s. For fluorescent labels, typical

acquisition times are 110 s, although the fastest reported so far is 5 ms (78).

Camera lag and interlacing must also be considered because they degrade the

time resolution (21a, 48, 88, 100).Colloidal gold, latex beads, and fluorescent particles have been used as labels.

Colloidal gold is a strong light scatterer that acts as a light sink rather than a light

source. Light is scattered out of the objective, so, after background subtraction

and contrast enhancement, the label appears darker than the surrounding image.

The diameter d of the gold particle is much less than the wavelength of light, so

the particle is a Rayleigh scatterer, for which the scattering d6. The minimum

detectable diameter is 15 nm and the typical diameter used is 3040 nm. Gold

particles are much stronger scatterers than organelles are, so the organelles are

almost invisible in bright-field microscopy (93). The use of gold labels isreviewed in References 21 and 23.

Fluorescent labels used include fluorescent microspheres, typically of diam-

eter 30100 nm (37, 46); phycobiliproteins (104); virus labeled with fluorescent

lipid analogs (2); low-density lipoprotein labeled with the carbocyanine lipid

analog diI (diI-LDL) (4, 42, 43); diI-LDL conjugated to immunoglobulin E

(IgE) (95); and tetramethylrhodamine conjugated to individual lipid molecules

(78, 79). Advantages and disadvantages of the labels are discussed in the

references.

There are several potential difficulties associated with different labels. First,most labels are large, so that drag from the interaction of the label with the

extracellular matrix may be significant (59, 60, 95). Second, labels are often

multivalent and can crosslink binding sites. Crosslinking lowers D through

hydrodynamic effects (1) and may trigger biological responses such as trans-

membrane signaling and interactions with the cytoskeleton. Furthermore, if

diffusion is restricted by corrals, crosslinking yields aggregates less likely to

cross corral walls (34, 68). Third, perturbations caused by antibody binding can

affect interactions of the labeled protein with other proteins (19, 52). Finally,

during a measurement, a particle may disappear as a result of moving out ofthe focal plane, endocytosis, detachment from the membrane, or photobleach-

ing (2). (For a detailed quantitative discussion of photobleaching of single

fluorophores, see 78.)

DATA ANALYSIS

The goal of SPT data analysis is to sort trajectories into various modes of

motion and to find the distribution of quantities characterizing the motion,

such as the diffusion coefficient, velocity, anomalous diffusion exponent, cor-ral size, and escape probability. The difficulty is that in a pure random walk

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the randomness yields trajectories that suggest other modes of motion. This

problem is made worse by the experimental limits on the duration of trajectories

measured (65, 72).

It is instructive to calibrate (or uncalibrate) ones intuition by writing a simple

random walk program and looking at a few dozen pure random walks. Onewill see apparent diffusion, directed motion, trapping, and transitions (66, 70)

because our nervous systems are wired to see patterns. But observation of mul-

tiple trajectories with the same apparently nonrandom behavior provides strong

evidence that the nonrandom behavior is real. The most striking experimental

examples are of directed motion (43) and motion among corrals (67).

In addition to the usual cellular, biochemical, and instrumental controls, it

is necessary to do controls for data analysis using a pure random walk as a

reference. The minimum test for a classification algorithm is to try it on pure

random walks of the appropriate number of time steps. A more rigorous testrequires both experiment and simulation. For example, consider the case of

corralled motion. First, the experimental corralled trajectories are identified by

some criterion. Then the criterion is applied to pure random walks to see how

many pure random walks are falsely classified as corralled, and to corralled

random walks to see how many corralled random walks are falsely identified as

free. Some corralled trajectories are necessarily rejected because their residence

times are by chance very low, so the average escape time is biased toward

higher escape times. To be able to do such tests, it is necessary to use some

algorithm to find quantities such as the initial slope, rather than finding themby eye.

When reporting classifications of trajectories based on some parameter, in-

clusion of a histogram of the parameter for the experimental data and pure

random walks (57, 67) is useful to show whether the classification is based on a

somewhat-arbitrary dividing line in a unimodal distribution or a minimum in a

multimodal distribution. Similarly, if multiple parameters are used, it is useful

to show them as a scatter plot (68).

To reduce the noise in an experimental trajectory, the data points within a

single trajectory are averaged, yielding the mean-square displacement (MSD)for that trajectory (65). The MSD for a given time lag can be defined as the

average over all independent pairs of points with that time lag (42), or all pairs

of points with that time lag. These averages are discussed in detail elsewhere

(MJ Saxton, manuscript submitted). Briefly, for time lags less than 14

of

the total number of points in the trajectory, the two averages agree, but the aver-

age over all points is less noisy. When the time lag is a substantial fraction of the

length of the trajectory, neither average is useful because there are simply not

enough data points, as shown by the formulas for the standard deviations (see

65). The short-range MSD is accurately determined, but the long-range MSDis noisy, yielding good short-range diffusion coefficients but highly scattered

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long-range ones. Averaging should not be done automatically because it may

obscure transitions between diffusive and nondiffusive segments of a trajectory.

The analytical forms of the curves of MSD versus time for the different

modes of motion (Figure 1) form the basis of various classification methods.

r2 = 4Dt normal diffusion (1)

r2 = 4Dt anomalous diffusion (2)

r2 = 4Dt + (V t)2 directed motion with diffusion (3)

r2

r2C

1 A1 exp4A2Dt

r2C

corralled motion (4)

In Equation 2, < 1, so strictly speaking this is anomalous subdiffusion (11,

36). In Equations 3 and 4, V is velocity, r2C is the corral size, and A1 and A2

are constants determined by the corral geometry. Equation 4 is based on the

first two terms of the exact series solutions for square corrals (57) and circular

corrals (70).

The probability density p(r, t)dr is the probability that a particle at the origin

at time zero is at position r at time t. For pure diffusion in two dimensions (65),

p(r, t)dr =1

4Dtexp(r2/4Dt)2r dr, (5)

and for diffusion with simultaneous flow along the x-axis with velocity V,

p(x ,y, t,V)d x d y =1

4Dtexp([(x V t)2 + y2]/4Dt)d x d y. (6)

For corralled motion, the probability density depends on the initial position in

the corral, and is complicated (57, 70).

Webb and collaborators (36, 95) assume that the probability density is the

standard two-dimensional form of Equation 5 but with a time-dependent diffu-

sion coefficient:

D = (1/4)t

1, (7)

or, equivalently,

r2 = 4Dt = t , (8)

with < 1. Diffusion is free at short times but slowed at longer times as the

effect of barriers becomes dominant. The physical basis for Equation 7 is the

idea of the membrane as a random array of continuously changing traps with

a distribution of energies so broad that there is no average residence time (36).

The continuous-time random walk (CTRW) model (63) gives the same formfor r2 at long times.

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Next, we summarize methods used to classify trajectories. Whatever the

method, the longer the run, the more reliable the classification unless the particle

changes its mode of motion.

Cherry and colleagues (2, 104) use the shape of the r2(t) curve to classify

trajectories. They calculate the experimental MSD and determine which analyt-ical expression yields the best fit. These workers also construct an experimental

probability density and fit sums of standard forms of p(r) to it.

Kusumi and colleagues (57) characterize the shape of the r2(t) curve in

terms of the relative deviation (RD). In effect, one draws a straight line through

the origin with the observed initial slope, which is known precisely and is not

affected significantly by nondiffusive motion. Then one extrapolates the MSD

to a prescribed time and takes RD to be the ratio of the observed MSD to the

extrapolated MSD. If RD > 1, then the motion is directed; if RD < 1, the

motion is confined. This approach reduces the shapes of the different curves ofFigure 1 to a single parameter. The distribution of RD can then be calculated for

a pure random walk and non-Brownian motion. The overlap of the distributions

is a measure of how well non-Brownian motion can be distinguished from pure

Brownian motion when using this parameter (57, 72).

Webb and collaborators (36, 95) use the anomalous diffusion exponent from

Equation 8. For each trajectory, log r2 is plotted versus log t, is found from

the initial slope, and the trajectory is classified according to .

The radius of gyration tensor is a well-known tool to characterize random

walks (66), which yields the asymmetry parameter a2 and the radius of gyrationR2gyr, a measure of the extent of the random walk. The joint distribution of R2gyr

and a2 may be used to classify trajectories (70). A related approach (93)

combines the observed D, the shape of the r2(t) curve, and the values of

R2gyr and a2 to sort trajectories into mobile, slowly diffusing, corralled, and

immobile.

APPLICATIONS

Results of SPT experiments, and the corresponding FRAP experiments whenavailable, are summarized in the tables. Table 1 includes artificial bilayers;

Table 2, lipids and GPI-linked (glycosylphosphatidylinositol) proteins in cells;

and Table 3, selected transmembrane proteins in cells. We believe the tables

include all the results to date for which both SPT and FRAP data are available.

Classification of Modes of Motion

FRAP measurements are generally interpreted as showing only mobile and

immobile fractions. SPT makes a much more detailed classification possible.

Practically all experimentalists report different modes of motion and transitions

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Table 1 Artificial bilayersa

Membrane Bilayer

component Label composition D(SPT)b D(FRAP) Ref.

Lipid analogs

TMR-POPE None POPC 140 23 77 13 78

Fi-PE 30 nm Au 87% egg PC 30 (MV) 133 33 58

Ab-Fl 13% Chol 70 (PV)

Biotin-PE 30 nm FM 80% egg PC 30 37

St 20% Chol

GPI-linked proteins

DAF (CD55) 30 nm FM 80% egg PC 25 37

St-biotin-Ab 20% Chol

FcRIIIB (CD16) same same 56 37

aAb, antibody; Chol, cholesterol; Fl, fluorescein; FM, fluorescent microsphere; GPI, glycosylphos-

phatidylinositol; MV, multivalent; PC, phosphatidylcholine; PE, phosphatidylethanolamine; POPC,palmitoyloleoyl PC; POPE, palmitoyloleoyl PE; PV, paucivalent; St, streptavidin; TMR, tetramethyl-rhodamine.bAll diffusion coefficients D in units 1010 cm2s1.

among the modes. Often, simple diffusion is observed in only a minority of

trajectories. Some data on modes of motion are given in the tables, but methods

of classification differ enough among laboratories that a table of modes of

motion is not useful.

Anomalous and Normal DiffusionOne of the most important results of SPT to date is the observation and mea-surement of anomalous diffusion in cell membranes. Anomalous diffusion can

be used as a probe of membrane organization. Furthermore, anomalous diffu-

sion implies slow diffusional mixing and therefore affects reaction rates in the

membrane (3).

What is the cause of this nonclassical behavior? In the most general terms,

anomalous diffusion results from a deviation from the central limit theorem,

resulting from pathologically broad distributions of jump times or jump lengths,

or strong correlations in diffusive motion (11). In cell membranes, anomalousdiffusion is most likely the result of both obstacles to diffusion and traps with

a distribution of binding energies or escape times.

For diffusion in the presence of random point obstacles (71), diffusion is

anomalous at short times and normal at long times: r2 t for t tC Rand r2 t for t tC R , where tC R is the crossover time and < 1. As the

obstacle concentration approaches the percolation threshold, decreases and

tC R increases, that is, diffusion becomes more anomalous for a longer time.

At the percolation threshold there is no crossover, and diffusion is anomalous

at all times because the percolation cluster is self-similar. For diffusion in the

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Table2

LipidsandGPI-linkedproteinsincellsa

Membrane

ApparentD(SPT)

component

Cell

Label

formo

bilefraction

D(FRAP)

Comments

Ref.

Lipidanalogs

Fi-PE

Fibroblasts

30nmAu

12

7

54

27

23%withD

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