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Cell Biology and Cell Biophysics Programme, European Molecular Biology Laboratory, 69117 Heidelberg, Germany
Correspondence: Address reprint requests to Matthias Weiss, Cell Biology and Cell Biophysics Programme, European Molecular Biology Laboratory, Meyerhofstr. 1, 69117 Heidelberg, Germany. Tel.: +49-6221-387-408; Fax: +49-6221-387-512; E-mail: mweiss{at}embl-heidelberg.de.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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1.5, which is determined by the anomality 
of the diffusional motion of the labeled particles, i.e., by the growth of their mean square displacement as (
x)2 
t. The fractality enforces an initial power-law behavior of the autocorrelation function and related quantities for small times. Using this information, we show by FCS that Golgi resident membrane proteins move subdiffusively in the endoplasmic reticulum and the Golgi apparatus in vivo. Based on Monte Carlo simulations for FCS on curved surfaces, we can rule out that the observed anomalous diffusion is a result of the complex topology of the membrane. The apparent mobility of particles as determined by FCS, however, is shown to depend crucially on the shape of the membrane and its motion in time. Due to this fact, the hydrodynamic radius of the tracked particles can be easily overestimated by an order of magnitude. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). For membrane proteins, diffusive motion enables segregation into subdomains and subsequently transport intermediates in the endoplasmic reticulum (ER) and Golgi apparatus (for reviews see Antonny and Shekman, 2001; Storrie and Nilsson, 2002). To better understand such processes, one wishes not only to determine the overall behavior of the proteins but also to gain detailed knowledge about their state. For example, is the protein of interest transiently part of an oligomeric complex (Nilsson et al., 1994), a lipid raft (Simons and Ikonen, 2000), and/or does it associate with the cytoskeleton (Yamaguchi and Fukuda, 1995)? Evidence for such interactions can be obtained by assessing the detailed diffusive behavior of the protein. Diffusion will be slower if associating transiently with the cytoskeleton and/or being partly in a lipid raft or a larger complex. It is likely that proteins experience more than one type of interaction in vivo, and if the interacting partners have a lower mobility than the protein in study, it will seem to take rests between times of free diffusional motion. As the resting times associated with these interactions are likely to vary widely, subdiffusive motion may be observed (Nagle, 1992; Saxton, 2001), i.e., the protein shows a time-dependent mobility and diffusion coefficient D(t).
Although it is likely that proteins experience multiple types of interactions resulting in subdiffusive behavior, it has proved difficult to determine this experimentally. Though with confocal bleaching techniques, i.e., fluorescence recovery after photobleaching (FRAP), a detection of anomalous diffusion is in principle possible (Saxton, 2001), this elusive information is often distorted by fluctuations in the recovery curve. Moreover, due to the complexity of the membrane morphology, anomalous diffusion can arise as a pure geometrical effect even if the particle diffuses normally (Sbalzarini, Mezzacasa, Helenius, and Koumoutsakos, unpublished). An alternative to FRAP is fluorescence correlation spectroscopy (FCS), which provides more local information and yields a higher temporal resolution at the level of single molecules (Rigler and Elson, 2001). However, even with FCS it is a challenge to detect and extract subdiffusive motion reliably (Schwille et al., 1999a,b).
Besides the issue of anomalous diffusion, the precise determination of mobilities is of high interest for the elucidation of biological questions, as it often serves as a key parameter for pattern formation (Murray, 1993). It was hypothesized, for example, that Golgi resident membrane proteins show complex formation upon reaching the Golgi (Nilsson et al., 1993, 1994). Similarly, unfolded membrane proteins in the ER were predicted to associate with a network of chaperones, which should lead to a lower mobility (Tatu and Helenius, 1997). Both questions have in part been addressed by FRAP in vivo (Cole et al., 1996; Nehls et al., 2000), and based on the apparent mobilities of the investigated proteins, it was argued that neither of the above hypotheses was correct. However, it remains open if FRAP or FCS can really assess the mobility in enough detail to definitely answer the posed questions.
Here, we provide evidence that anomalous diffusion manifests itself as fractal fluctuations of the fluorescence F(t), i.e., the fractal dimension D0 of F(t) depends on the anomality of the diffusion. As a consequence, the autocorrelation function of F(t) and the variance of fluorescence increments show a power-law behavior for small times. We confirm this by computer simulations and employ this knowledge to evaluate experimental FCS curves of membrane proteins in the ER and Golgi apparatus, in vivo. We find that all tested proteins move subdiffusively, but have different degrees of anomality. Secondly, we show that the shape of the membrane does not induce subdiffusive behavior even if the curvature changes with time, but it does affect the apparent mobility. In fact, the uncertainty in the mobility is typically big enough to prevent a monomeric membrane protein to be distinguished from a cluster consisting of several hundred monomers.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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 | (1) |
5 - 10 (Rigler and Elson, 2001). The density n(r,t) of labeled particles at a point r in the focus fluctuates as particles are subject to diffusion, yielding the time-averaged autocorrelation function
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(r,r',
) of the diffusion equation, i.e., the propagator of the density n(r,t) and the integrations run over all times and the whole space, respectively. Separating the fluctuations and the average density by 
and assuming the fluctuations to have zero mean and to be uncorrelated at different spatial positions at any time point yields: 
As the measurable signal is the total fluorescence in the confocal volume 
one obtains from that and Eq. 1 the autocorrelation function of the fluorescence signal
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Here, both integrals are taken over the whole space and A is a constant that depends on 
I0, and properties of the fluorophore. As we are only interested in the decay of the correlation, we define by
 | (2) |
.
We note that the decay of C() is essentially determined by the Greens function of the diffusion equation. Denoting by gij the metric of the (curved) manifold of diffusion in Cartesian coordinates, its inverse by gij and letting 
the diffusion equation reads in general
 | (3) |
). The Greens function (r,r',t) satisfies Eq. 3 with the constraint (r,r',t = 0) = 
(r - r') and can be calculated analytically only in rare cases, e.g., for diffusion on the xy plane perpendicular to the optical axis. In that case the metric is particularly simple (gij = ij, i, j = 1,2) and one obtains from Eq. 3 the familiar two-dimensional diffusion equation with the Greens function 
With that, the integrals in Eq. 2 can be solved analytically, yielding the well-known autocorrelation function
 | (4) |
is the diffusive time and B is a constant that, in general, includes the average number of molecules in the focus and properties of the fluorophore. In case of two different, noninteracting kinds of particles labeled with the same fluorophore but having diffusive times 
and fractions f, 1 - f, respectively, the total correlation curve is simply the sum of two individual curves (Schwille et al., 1999a):
 | (5) |
Monte Carlo simulations of anomalous and constrained diffusion
In all simulations, we have fixed the characteristic radius of the confocal spot to r0 = 0.3 µm in the xy plane and chose the laser intensity to be I0 = 1. The radius of the confocal volume in z direction was chosen to be fivefold bigger (S = 5), which is a typical value (Rigler and Elson, 2001). To test the fractality of F(t), we initially distributed 100 particles randomly on a two-dimensional plane (edge length L = 2.1 µm) with periodic boundaries, perpendicular to the optical axis. The erratic motion was simulated using the forward integration of the Langevin equation, i.e., x(t + dt) = x(t) + 
(dt), with time increments dt = 1 µs. As a model for subdiffusive motion, we have chosen to calculate the spatial increments (dt) in x and y direction via the Weierstrass-Mandelbrot function (Berry and Lewis, 1980; Saxton, 2001)
 | (6) |
n are random phases in the interval [0,2
], 
> 1 is an irrational number, t* = 2t/tmax is connected to the maximum length tmax of the desired time series, and H = /2 is the Hurst coefficient leading to a growth of the particles' mean square displacement as (x)2 t. In accordance with Saxton (2001), we chose 
and restricted the sum to the terms n = -8, ..., 48. The increments (dt) = W(t + dt) - W(t) were scaled to match the desired values of the transport coefficient 
. For the chosen = 1,0.7,0.5 and = 57/s,50/s0.7,39/s0.5, we took 10 time series of length tmax = 5 s for averaging.
Fractal analysis was performed by using a box-counting algorithm, i.e., the curve F(t) is covered with squares of length and the number of filled boxes 
() is counted. This quantity shows a scaling 
In contrast, C() and v() were calculated directly from F(t). While v() in the simulations was monotonically increasing with , the experimental curve had an offset v0 = v( 0) > 0 due to the stochastic nature of the emission of the fluorophore. Consequently we subtracted this offset and analyzed v() - v0.
Monte Carlo simulations of diffusion on surfaces with nonzero curvature, defined by (r) = 0, were performed using the algorithm proposed in (Ho
yst et al., 1999). In brief, we calculated the local Cartesian frame of the tangential surface (r - r0)·
(r0) = 0 at the position r0 of every particle. We then drew Gaussian random numbers with variance 2Dt for the in-plane stochastic movement in each direction of the local Cartesian frame. The resulting vector r1 was projected back to the surface = 0 yielding the new position 
For sufficiently small t, the erratic movement of the particles is bound to the surface (Hoyst et al., 1999). In all simulations we used I0 = 1, r0 = 0.3 µm, 100 particles, t = 10-7 s, D = 1 µm2/s, and S = 5.
Cell culture, fluorescence microscopy, and FCS
Stable HeLa cell lines were constructed by transfection with plasmid DNA encoding the trans-membrane proteins GalNAc-T2 or p24ß1 fused to green fluorescent protein (GFP) and yellow fluorescent protein (YFP), respectively, and selected for using G418 as described previously (Storrie et al., 1998). Cells were grown in Dulbecco's modified Eagle's medium with 10% fetal calf serum, 100 µg/ml penicillin, 100 mg/ml streptomycin, and 10 mM glutamine (Gibco, Eggenstein, Germany). HeLa cells expressing GalT(GFP) were obtained by microinjecting purified pGalT plasmid (concentration 50 ng/ml) into cell nuclei using an Eppendorf microinjection system (Eppendorf, Hamburg, Germany) and Cascade blue dextran (Molecular Probes, Eugene, OR) as a coinjection marker.
FCS measurements were carried out with a commercial instrument (ConfoCor II, Carl Zeiss, Jena, Germany), combining a laser scanning and an FCS unit. The used objective was an Apochromat 40x/1.2 W. With this setup, a spot on a previously scanned image of a cell could be selected for the FCS measurement. The spatial resolution is 0.5 µm, corresponding to an average spot size of 0.3 µm. GFP-tagged proteins were illuminated at 488 nm and detected with a bandpass filter (505–550 nm). To reduce the unavoidable deviations of the optical volume from a Gaussian form, we used very low laser intensities in the range 40–100 µW. As can be seen from Fig. 7 a, in this range the approximation of a Gaussian confocal volume is well in agreement with previous reports for the ConfoCor II (T. Jankowski and R. Janka, 2001). For higher laser powers, however, the deviations become much stronger (Fig. 7 b), in agreement with Hess and Webb (2002). More details on this issue are given in the Discussion. The pinhole for all shown measurements was 1 Airy unit (70 µm) unless stated otherwise. Calibration of the diffusional timescale was done with GFP in water (D 87 µm2/s) (Wachsmuth et al., 2000). For the measurements on GFP-tagged proteins, 10–20 time series of 10 s were recorded with a time resolution of 1 µs and then superimposed for fitting. Measurements on the Golgi were preceded by bleaching to decrease the fluorescence below the saturation value of the detectors.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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1 fl. Anomalous diffusion implies that the particles' mean square displacement grows asymptotically as (x)2 = 4t, 
1, and the autocorrelation function for particles moving on a plane perpendicular to the optical axis then reads in general
 | (7) |
= 1). The transport coefficient has the dimension of an area per fractional time (µm2/s), and one can define a time-dependent diffusion coefficient by D(t) = t-1. This expression converges to a constant 
for normal diffusion ( 1), where D, defined by C() = B/2, is the diffusive time. Fitting the experimental FCS curve with Eq. 7 therefore yields the local transport coefficient and the degree of anomality .
To overcome the reported ambiguity that experimental FCS curves can be fitted successfully with Eq. 7 and expressions that describe coexisting populations of proteins with different mobilities (Schwille et al., 1999a,b) (cf. Eq. 5), we took a closer look at the fluctuations of the fluorescence time series F(t), i.e., its basic statistical properties. Restricting the correlation analysis to the regime 
one obtains from Eq. 7 C() const. - . This scaling is very similar to the characteristic behavior found for a certain class of stochastic processes known as fractional Brownian motion (Mandelbrot and Ness, 1968). These processes are characterized by the above asymptotic algebraic autocorrelation decay, a growth of the variance of (fluorescence) increments v() = 
(F(t) - F(t + ))2
t and a fractal dimension D0 = 2 - /2 of the curve F(t). These signatures of fractional Brownian motion have been found and characterized in detail, e.g., in conductance fluctuations of mesoscopic samples in solid-state physics (Hufnagel et al., 2001; Maspero et al., 2000). From the analogy of fractional Brownian motion and the small- limit of Eq. 7, we conclude that the fluorescence time series F(t) taken in FCS measurements should exhibit a fractal dimension D0 = 2 - /2 on short timescales and we anticipate an increase v() = (F(t) - F(t + ))2t for the variance of the fluorescence increments. In fact, the scaling v() t is intimately related to the initial decay of C() via v() = const. - C() and therefore it is a sensitive measure for the anomality when restricting the analysis to small times. A fractal analysis of F(t) therefore should provide the desired informations D0 and about the diffusive process without having to fit the entire autocorrelation function C().
We have tested our expectations quantitatively with the help of Monte Carlo simulations using different levels of subdiffusion (see Methods). We simulated (sub)diffusive motion of particles on a plane for = 1, 0.7, 0.5 and monitored the fluorescence F(t) arising from a Gaussian intensity profile. A fractal analysis of this signal confirmed the fractality of F(t) (D0 = 2 - /2), the relation v() , whereas C() was perfectly described by Eq. 7 (see Fig. 1).
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determines the fractal dimension D0, our approach should thus be capable of reliably extracting for one-, two-, and three-dimensional motion with an arbitrary number of different components.
Anomalous diffusion of membrane proteins in vivo
We next tested the fractal analysis as a method to investigate anomalous diffusion of membrane proteins in vivo. Three Golgi resident enzymes tagged with GFP, N-acetylgalactosyltransferase 2 (GalNAc-T2(GFP)), 1,4ß-galactosyltransferase GalT(GFP), and p24ß1(YFP) were monitored both in the ER and in the Golgi. As only 5–10% of these proteins localize to the ER (Storrie et al., 1998) (see also Fig. 2, a–c), we could be certain to meet the low-fluorescence situation typically anticipated in FCS measurements. From the offset of the autocorrelation curves in the ER, we estimated the average number of proteins in the focus in all cases to be <30. For the Golgi, we performed a prebleach before the measurement to avoid saturation of the FCS detector.
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) for GalNAc-T2 is described equally well by Eq. 7 for anomalous diffusion and by an expression describing the coexistence of two populations with different mobilities (Methods, Eq. 5). However, these fits describe entirely different behaviors: whereas the first predicts a time-dependent diffusion coefficient D(t) = 0.2/t0.25 µm2/s0.75 as = 0.75, the second predicts two GalNAc-T2 populations having diffusion constants D1 = 0.3 µm2/s and D2 = 7 µm2/s. Similar to earlier reports (Schwille et al., 1999a,b), one cannot determine which of the two scenarios is actually underlying the experimental data.
Due to the low level of fluorescence, the scaling of () did not extend over a broad enough range in to allow for a reliable estimate of D0 and . Therefore, the scaling of v() for small was the only way to test for anomalous diffusion. As will become clear in the next section, it provides nevertheless a more valuable tool to determine the anomality than fitting the entire function C(). Indeed, we found a scaling v() , = 0.7 ± 0.05 (Fig. 2, inset) that agrees well with the fitting of the entire C() with Eq. 7 over four orders of magnitude in . We confirmed this result for different cells and various ER sites. From this we conclude that GalNAc-T2 actually moves subdiffusively in the ER.
A close look at the diffusion constant of proteins in membranes (Saffman and Delbrück, 1975), i.e., D = kBT(ln{h
m/cR} - 0.5772)/(4mh) supports this conclusion: estimating the radius of the membrane spanning domain of GalNAc-T2 to be R 1 nm (slightly bigger than the size of a single -helix), the thickness of the membrane to be h 6 nm and assuming the membranous and cytosolic viscosities to be m 0.15 kg m/s and c 0.003 kg m/s, the ratio D2/D1 = 23.5 obtained from Fig. 2 d by fitting with a multiple-component expression implies a hydrodynamic radius R 135 nm of the second component. In fact, this particle then had to consist of more than 104 GalNAc-T2 monomers, which appears somewhat large.
To rule out that the observed anomalous behavior was a special feature of GalNAc-T2, we also tested the ER pool of GalT and p24ß1, two integral membrane proteins that localize preferentially to the late and early Golgi apparatus, respectively (Fig. 2, b and c). Again we found anomalous diffusion for these proteins (Fig. 2, d and e). Whereas the anomality for GalT ( = 0.5 ± 0.05) appeared to be even stronger than for GalNAc-T2, p24ß1 showed a slightly less subdiffusive behavior ( = 0.8 ± 0.05). We next monitored the mobility of GalNAc-T2 and p24ß1 in the Golgi apparatus. As shown in Fig. 3, the same degrees of subdiffusiveness were also observed in the Golgi.
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) plays a major role. Inactivating the COPII machinery by a Sar1dn GDP-restricted mutant that blocks export from the ER also did not alter the obtained results. We speculate that the most likely explanation for the observed subdiffusion is a continuous time random walk, i.e., a random walk with rests due to transitional bindings in between. Dicing the resting times according to an algebraic distribution, one can obtain any value of 0 < < 1 (Nagle, 1992; Saxton, 2001). Although obstacles on the membrane will also contribute to subdiffusive motion (Saxton, 2001), this can reduce the anomality even at the percolation threshold only to 0.7, which is bigger than the observed value = 0.5 for GalT. The continuous time random walk may be caused by transient bindings to larger proteins, e.g., the UDP-glucose transporter or specialized protein-lipid domains (Simons and Ikonen, 2000).
Influence of the membrane shape on the apparent mobility
Whereas we could confirm the degree of anomality for integral membrane proteins under different conditions, the transport coefficient showed considerable variations. We reasoned that this was most probably caused by the local morphology of the membrane. The effect of geometrical constraints on FCS has so far not been a point of major considerations. Only the special case of three-dimensional diffusion between fixed, reflecting boundaries has been studied in some detail (Gennerich and Schild, 2000). For FCS on membranes, however, most experimental data are still evaluated with the simple expression Eq. 4. In the following, we restrict ourselves to normal diffusion and we first turn to the easiest nontrivial extension of a plane membrane perpendicular to the optical axis (Eq. 4), i.e., a tilted plane. Without loss of generality, we assume that the tilted plane emerges by rotating the xy plane by an angle 
around the y axis (a shift with respect to the focus will only affect the prefactor B in which we are not interested). The metric for this tilted plane in Cartesian coordinates is given by g11 = 1 + tan2(), g22 = 1, g12 = g21 = 0, from which one obtains the autocorrelation function
 | (8) |
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0. Some representative examples of Eq. 8 for various are shown in Fig. 4 a. In real applications, however, one may not know if and how much a membrane is actually tilted. Fitting C() then naively with Eq. 4 results in an overestimation of D, which may be as high as 2.5-fold (Fig. 4 a, inset).
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may be a function of time. We have simulated this by changing periodically by a factor cos(
t) with = 0.2, 2, 20 Hz (Fig. 4 b). The obtained curves are well described by the angle-averaged expression of Eq. 8, i.e.,
 | (9) |
D.
We speculated that diffusion on curved surfaces could yield even higher overestimations of D, when fitting naively with Eq. 4. We concentrated on hyperbolic and parabolic surfaces as the easiest examples of a 2D surface with nonzero curvature defined by
 | (10) |
= +1 and = -1, respectively. In these cases all components of the metric tensor in Cartesian coordinates are algebraic functions of x,y and thus solving Eq. 3 to obtain (r,r',t) becomes very complicated. As we have not been able to derive C() analytically, we performed Monte Carlo simulations to gain some insight in the behavior of the correlation decay. For the sake of simplicity, we have set a = b and investigated the autocorrelation function for various a. In Fig. 5 a, the result for the hyperbolic surface is shown in comparison to C() derived from Eq. 4 when inserting the mobility used in the simulations. Obviously an overestimation of D up to a factor 4.5 with decreasing a is observed and the shape of the autocorrelation function seems to be stretched. Therefore, it was tempting to compare the curves to Eq. 7, which yields a reasonable fit. It thus seems as if the degree of subdiffusiveness depended on a and tends to unity for a >> r0. A fractal analysis of the fluorescence, i.e., using a box-counting algorithm (Maspero et al., 2000) and inspecting the small time limit of v(), does not yield any hint that anomalous diffusion produces the observed C(), as anticipated from the definition of the simulation. This is a nice example, where the fractal analysis, and in particular the behavior of v() for small times, can decide whether anomalous diffusion is observed or if it is rather an effect induced by geometry.
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in Eq. 10 to perform a random walk in the interval (-1, 1) with different "diffusion" constants, thereby interpolating between hyperbolic and parabolic surfaces. The resulting C() is shown in Fig. 5 c. The curves not only decay slower than the simple expression Eq. 4, but also show a different decay behavior, which is reminiscent of the shapes observed for the hyperbolic manifolds. Again, a fractal analysis did not show any traces of anomalous diffusion, whereas C() on the whole was well described by Eq. 7.
As another, more complicated surface with a different topology, we studied the G periodic nodal surface, defined by
 | (11) |
x/d0, Y = 2y/d0, Z = 2z/d0, and d0 is the characteristic length between two nodes. This surface reminds in its topology of the ER of living cells (see Fig. 6, inset) and we therefore used it as a model for this membrane compartment. The FCS curve and its fractal analysis for this surface showed normal diffusive behavior as anticipated from the definition of the simulation (Fig. 6). However, the autocorrelation curve decays slower than expected for particles moving on a plane membrane, though the underestimation of the mobility is less strong than in the previous cases.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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of anomalous diffusion. In particular, the behavior of the autocorrelation function C() and the variance of fluorescence increments v() is governed by power laws in the limit of small , where the exponent is simply given by . Using this information, we could determine that GalNAc-T2(GFP), GalT(GFP), and p24ß1(YFP) move subdiffusively in the ER and the Golgi apparatus with different degrees of anomality . This seemed to be an inherent property of the protein rather than an effect of the particular environment. Using a simulation approach, we could rule out that the membrane shape induces anomalous diffusion, although it can change the apparent mobility by up to an order of magnitude.
Although anomalous diffusion could also, in principle, be detected by the asymptotic power-law decay C() , 
when displaying log{C()} versus log{}, this approach is usually not feasible as it requires very long time series. About 3–4 orders of magnitude beyond 
with sufficient statistics are needed to enable this, i.e., typically a time series F(t) of length tmax > 1000 s is required for statistical significance in the regime 100 s. While for giant liposomes these times may still be feasible, they are prohibitively long for in vivo applications when membranes are subject to movement and reformation, which perturbs the anticipated information. Also, the shape of the membrane can considerably influence the shape of C() and mimic anomalous diffusion, although actually normal diffusion is observed. In contrast, our approach of analyzing the fractality of F(t) on short timescales allows us to neglect the membrane curvature and yields a way to estimate if and to what degree anomalous diffusion plays a role in the FCS signal, even when tmax 10 s. Our approach to estimate requires a window of at least 1–2 orders of magnitude in to determine the involved power laws. We would like to stress that the range of fitting is very important for determining the proper value of . An upper boundary for the fitting range is given by the restriction that the scaling v() can only be expected to hold true as long as 
The lower boundary is less well defined, but is essentially dictated by the conversion of fluorophores to the triplet state on time scales 10 µs. Our method is only applicable when the half time of C() is at least 1–2 orders of magnitude bigger than this triplet time and if the fraction of molecules in the triplet state is low. As the fraction of dye molecules in the triplet state rises with the intensity of the laser light (Widengren, 2001), it is essential to use low powers. As anomalous diffusion typically yields 
for 1 ms in the nucleus of cells (Wachsmuth et al., 2000) and 10 ms on membranes (this study), one has under these conditions 1–2 orders of magnitude for applying the fractal analysis. Our experimental data confirm this reasoning.
A possible source of error in our above analysis is the recently reported deviation of the confocal volume from the theoretically assumed Gaussian form (Hess and Webb, 2002). These deviations can lead to artifacts during the data evaluation when simply assuming a Gaussian volume. For example, the correlation function can mimic a two-component system even if only a single species is present. Likewise, C() could also be fitted well by anomalous expressions such as Eq. 7 yielding 0.9 (Hess and Webb, 2002), which is, however, considerably higher than the values found here (Fig. 3 c). To address concerns that our confocal volume is non-Gaussian, we have monitored the free diffusion of fluorescein isothiocyanate-labeled immunoglobulins (IgG) in buffer. Due to the high molecular weight of IgG, the diffusional decay of C() is well separated from the triplet contribution on scales << 10 µs (Widengren, 2001). We tested various laser powers and pinhole sizes, fitted C() with an expression for diffusional motion in three dimensions (Rigler and Elson, 2001), and calculated the residuals. A representative curve for the conditions that were used in the live cell measurements is shown in Fig. 7 a. The extremely low residuals indicate that the confocal volume indeed is very well approximated by a Gaussian form. We therefore conclude that the deviations of the confocal volume from a Gaussian form do not play a major role in our measurements. We note that for higher laser powers, the residuals grow considerably (Fig. 7 b). We therefore recommend the use of low laser intensities also for this reason.
The observed deviations from a Gaussian confocal volume also limit the range over which the influence of the triplet kinetics can be studied while still keeping the confocal volume Gaussian. However, as the value of stayed constant in all live cell measurements with laser intensities 40 µW I 100 µW (I changed from cell to cell depending on the expression level), the triplet kinetics does not seem to play a major role for our analysis. Also changing the pinhole did not affect (see Fig. 7 c for an example).
We rule out the possibility that the anomalous decaying C() could be due to multiple components with a peculiar distribution of diffusive times. Although theoretically possible, this problem is unlikely to play a role in this study: there exists a minimum diffusive time for an integral membrane protein that can be estimated via the Saffmann-Delbrück equation, i.e., D 2 µm2/s 
D 3 ms. This represents a lower cutoff for the distribution of diffusive times, and if the motion of the proteins would not be anomalous, one should observe v() for << 3 ms. As the scaling v() is also observed on scales << 3 ms (see Figs. 2 and 3), it is very unlikely that we have monitored multiple components with a peculiar distribution of diffusive times.
Finally, we would like to comment on the influence of the membrane shape on the estimate for the mobility of membrane proteins. As reported here, the apparent mobility can be changed by an order of magnitude depending on the local shape of the membrane. Similar observations have been made for FRAP applications (Aizenbud and Gershon, 1982). One may wonder if this uncertainty is in any way crucial. In fact, taking the Saffmann-Delbrück equation for the diffusion coefficient of membrane proteins (Saffman and Delbrück, 1975), one can infer that a factor of two in the mobility can correspond to a difference in size of the object by a factor of 10. If it is a cluster of proteins, this would correspond then to an assembly of 200 monomeric proteins. In light of our findings, it will be interesting to revisit previous studies (Cole et al., 1996; Nehls et al., 2000) monitoring membrane protein dynamics in the ER and in the Golgi apparatus.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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M.W. acknowledges financial support by a European Molecular Biology Organization long-term fellowship.
Submitted on October 30, 2002; accepted for publication February 11, 2003.
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(dashed lines). (c) The variance of the fluorescence increments (full lines) shows the expected behavior v(
of the diffusive time 
ms as a function of 
[-
All times 
