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* MEMPHYS-Center for Biomembrane Physics, Physics Department, University of Southern Denmark, Campusvej 55, Odense M, Denmark; and Department of Medical Biochemistry, Göteborg University, Medicinaregatan 9A, Göteborg, Sweden
Correspondence: Address reprint requests to Matthias Weiss, MEMPHYS-Center for Biomembrane Physics, Physics Department, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. Tel.: 45-6550-3686; E-mail: mweiss{at}memphys.sdu.dk.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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100 nm) the endoplasmic reticulum (ER) imposes a random reticular network (Marsh et al., 2001
) together with the cytoskeletal elements, such as microtubuli and actin filaments. Together, these yield a higher order structure of the cytoplasm (see, for example, Alberts et al., 1994 for a more detailed introduction). As a consequence, diffusional movement of particles, such as macromolecules, can be obstructed. In fact, it has been reported that the diffusional mobility in the cytoplasm strongly decreases with an increasing radius of the tracked particle, leaving particles with a radius >25–30 nm immobile (Luby-Phelps et al., 1986, 1987; Seksek et al., 1997; Arrio-Dupont et al., 2000). Extensive computer simulations also have shown that the molecular mobility is reduced when a particle diffuses in a maze-like environment (Saxton, 1993): When increasing the concentration c of obstacles in the maze, the tracer particles appeared to diffuse slower and slower until complete immobilization occurred beyond a certain value, c*. Interestingly, when approaching c* the characteristics of the diffusional motion changed dramatically. The mean square displacement v(t) of the monitored particles did no longer grow linearly in time but, rather, showed a power law v(t) t
with < 1. This kind of diffusion is known as anomalous subdiffusion and has been found in many different contexts; e.g., for the movement of lipids on model membranes (Schutz et al., 1997), integral membrane proteins on organellar membranes (Weiss et al., 2003) and proteins in the nucleoplasm (Wachsmuth et al., 2000), solute transport in porous media (Drazer and Zanette, 1999), and the translocation of polymers (Metzler and Klafter, 2003; Kantor and Kardar, 2004).
In the case of obstructed diffusion, the emergence of a transitional subdiffusive regime is observed when the concentration of obstacles is increased. This transient subdiffusive behavior collapses back to normal diffusion after a timescale T which diverges in the limit c 
c*. At c = c* (the so-called percolation threshold), subdiffusion is observed on all timescales. Whereas T grows with increasing obstacle concentration, the (transient) anomality parameter decreases concomitantly from unity to a finite value * at c*, which is given by * 
0.697 and * 0.526 for two- and three-dimensional environments, respectively (Havlin and Ben-Avraham, 1987; Bouchaud and Georges, 1990). These values were obtained for continuum percolation in a "Swiss-cheese" model (see Havlin and Ben-Avraham, 1987 for details) and presumably represent the best approximation to the actual values in nature. However, other mechanisms can also lead to anomalous subdiffusion where the entire range 0 < < 1 may be observed (see, for example, Bouchaud and Georges, 1990; Metzler and Klafter, 2000). Regardless of its microscopic origin, anomalous subdiffusion has been shown to strongly influence the formation of spatiotemporal patterns (Weiss, 2003) as well as kinetic rates (Saxton, 2002) and the time course of enzymatic reactions (Berry, 2002).
When neglecting the higher-order structuring of the cytoplasm by cytoskeletal elements and membranes, one could anticipate from the above that one deals with an unstructured aqueous solution in which normal diffusion should be observed. Yet, the assumption of the cytoplasm as being a homogenous viscous solution is somewhat misleading as differently sized proteins, lipids, and sugars constitute up to 40% of the cytoplasmic volume (Fulton, 1982). This phenomenon is commonly referred to as molecular crowding and has recently received increased attention (Ellis and Minton, 2003; Rivas et al., 2004) since, for example, enzymatic reactions and protein folding appear to be strongly affected by the crowdedness (for reviews see Ellis, 2001; Hall and Minton, 2003). Also, crowding seems to contribute significantly to the high viscosity of the cytoplasm which has been determined to be three- to fourfold higher than that of water (Verkman, 2002; Elsner et al., 2003). Despite the increased interest in the phenomena associated with molecular crowding, the term "crowdedness" so far has been used without a quantitative definition of what it actually means. In other words, we lack a definition of a quantity which summarizes how crowded an environment really is and also states in which primary physical property of the heterogeneous fluid the crowdedness is manifested. As basic criterion, a quantitative measure of crowdedness should be independent of influences imposed by the cytoskeletal and membrane obstacles discussed above. Rather, it should reflect a basic and unambiguous physical quantity which can be assigned to the highly, yet heterogeneous, concentrated protein/sugar solution called cytoplasm.
Here we utilize fluorescence correlation spectroscopy (FCS) to show that inert tracer particles show anomalous subdiffusion in the cytoplasm of living cells over a wide range of particle sizes. This behavior is found to occur irrespective of the stage of the cell cycle or the presence of ER membrane structures and cytoskeletal scaffolds. Using computer simulations, we demonstrate that this effect most likely arises due to molecular crowding, e.g., diffusing particles are scattered by nearby particles due to excluded-volume interactions. We verify our hypothesis in vitro by determining the degree of anomalous diffusion of tracer particles in highly concentrated dextran solutions.
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MATERIALS AND METHODS |
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for details). Experiments using mitotic cells were accomplished by arresting HeLa cells in the metaphase of mitosis by incubating them for 16 h in the presence of 100 nM nocodazole (Sigma Chemical, St. Louis, MO) (Zieve et al., 1980). For subcellular fractionation, HeLa cells were scraped off the culture dish and collected by centrifugation (500g, 5 min.). Cells were washed with phosphate-buffered saline (PBS) twice and once with homogenization buffer. The homogenization buffer consisted of 20 mM HEPES-KOH (pH 7.4), 1 mM DTT (both Biomol, Hamburg, Germany), 250 mM sucrose (USB, Cleveland, OH), 1 mM EDTA (Merck, Hamburg, Germany), plus protease inhibitors (1 µg/ml aprotinin, 1 µg/ml leupeptin, 1 µg/ml pepstatin, 1 µg/ml antipain, 1 mM Benzamidine-HCl, 40 µg/ml phenylmethylsulfonyl fluoride). Cell pellets were resuspended in 4 volumes of homogenization buffer in the presence of protease inhibitors and homogenized using a ball-bearing homogenizer (10 passages with a 16 µm clearing). The homogenate was then centrifuged sequentially at 103g (P1), 104g (P10), and at 105g (P100), retaining the supernatant at each subsequent centrifugation step. The final 105g supernatant (S100) was boiled in equal volume sample buffer and various amounts (0.1–10 µg) of protein were resolved on a 12.5% SDS-polyacrylamide gel. Protein bands were visualized by Coomassie Brilliant blue G250 (Merck, Darmstadt, Germany).
Fluorescence microscopy and FCS
FCS measurements were carried out on a LSM510/ConfoCor 2 (Carl Zeiss, Jena, Germany) using a 488-nm laser line for illumination. The fluorescence was detected with a bandpass filter (505–550 nm) and the objective (Apochromat 40x/1.2 W) was heated to 37°C using an objective heater (Bioptechs, Butler, PA). The pinhole for all shown measurements was 1 Airy unit. We verified that for free diffusion in water, the autocorrelation function of the fluorescence was well fitted by Eq. 1 with = 1. Thus, our analysis does not suffer from deviations of the confocal volume from a three-dimensional Gaussian point-spread function (see also discussions in Hess and Webb, 2002; Weiss et al., 2003). For each cell and condition, at least 30 fluorescence time series of 10 s duration were recorded, autocorrelated, and superimposed for fitting with XMGRACE (see http://plasma-gate.weizmann.ac.il/Grace/).
Autocorrelation times 
D were translated into apparent hydrodynamic radii by comparison with green fluorescent protein (EGFP, Molecular Probes) in PBS: From the diffusion coefficient D 85 µm2/s of GFP in buffer (Terry et al., 1995) and the determined diffusive time D = 130 µs, we obtained via the Einstein-Stokes equation D = kBT/(6
r) a mean radius r = 2.6 nm for GFP (kBT 4.3 x 10–21 J is the thermal energy and 10–3 kg/(m x s) is the viscosity of water). This value agrees well with the dimensions derived from the crystal structure of GFP (Yang et al., 1996).
Fitting anomalous diffusion
To determine if the experimentally observed autocorrelation function C() is governed by anomalous subdiffusion one has to generalize the well-known expression for the autocorrelation decay due to normal diffusion. Knowing the illumination profile (which is usually approximated by a three-dimensional Gaussian), this task is essentially done when the propagator 
of the density of the (sub)diffusing particles is known. This function simply tells the probability to find a particle at position 
after a time when it was initially at position 
For normal diffusion is simply a Gaussian which satisfies the diffusion equation and it is easy to derive the appropriate expression for C() (for details see, for example, Hess and Webb, 2002; Weiss et al., 2003). In contrast, the propagator for subdiffusion is somewhat more difficult to obtain. Bearing in mind that subdiffusion is commonly defined via the asymptotic power-law increase of the mean square displacement v(t) t ( < 1), a straight-forward (yet approximative) approach to determine 
is to assume a time-dependent diffusion coefficient D(t) = 
t–1 so that v(t) = D(t) x t. Clearly, this interpretation is problematic for small times as D(t) diverges for t 0. Yet, assuming that one still can use this approximation for all times, one obtains the propagator
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Here, is the degree of the anomalous subdiffusion, and D is the diffusive time which is related to the diffusion coefficient D and the width r0 of the focus as 
for = 1. The parameter S considers the unavoidable extension of the confocal volume along the optical axis, whereas f, T are the triplet fraction and time, respectively, which take care of the photophysics on short timescales.
The fitting function Eq. 1 has been used previously to determine anomalous subdiffusion in FCS experiments (Schwille et al., 1999; Wachsmuth et al., 2000; Weiss et al., 2003) and the very same approach served as a starting point to derive fitting functions for quantitative photobleaching experiments (Feder et al., 1996; Saxton, 2001). However, the outlined strategy appears somewhat questionable due to the divergence of the time-dependent diffusion coefficient on short timescales. A mathematically correct treatment of the problem therefore has to employ a fractional Fokker-Planck equation (FFPE), i.e., a sophisticated extension of the normal diffusion equation. For the FFPE one can analytically calculate the propagator in terms of Fox functions for all < 1 (see Metzler and Klafter, 2000). From this, one could derive C() analytically. However, the emerging function only has a limited value for a later fitting procedure as its complexity severely hampers the fitting to experimental data. We therefore have chosen a different approach: Using the series expansions of the propagator (cf. Metzler and Klafter, 2000), we calculated numerically the propagator and the resulting correlation function. We then fitted these curves with Eq. 1 (fixing the triplet fraction to f = 0) to test if the obtained value fit corresponds to the value FPE imposed in the FFPE. In all cases, Eq. 1 yielded a good fit to the C() as obtained from the FFPE (see Fig. 1 for a representative example). The anomality degrees fit and FPE on the other hand were slightly different (Fig. 1, inset) and a linear regression yielded fit = 1.1 x FPE – 0.12. In the range 0.5 
1 the deviations between Eq. 1 and the FFPE is therefore <10% which is within the accuracy of the experimental data. In view of this and due to its much simpler use in the fitting procedure, we have chosen to always use Eq. 1 for fitting.
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): Each particle k experienced a (repulsive) force 
from a neighboring molecule i along the vector 
pointing from particle k to particle i. Here, d measures the distance between the particles i, k, minus the radii ri, rk of the two particles. For d > rc the particles do not meet and thus 
Besides this excluded volume interaction, all particles were also subject to thermal noise, i.e., for each time step 
t the new position emerged from the old one via the (overdamped) Langevin equation 
Here, 
is Gaussian random number with variance 2Dit and the friction of the particle is assumed to be given by Stoke's formula (
i = 6ri) from which one also obtains the diffusion coefficient via Di = kBT/i. The radii were calculated from the imposed molecular mass mi via the empiric formula ri = (8mi/50)1/3 nm. This relation has been derived by considering that BSA (m = 66 kDa) is approximately globular and has an apparent radius of 2 nm. The distribution p(m) of molecular weights m was taken to be either a Poissonian or uniform (see main text), and a upper cutoff at m = 1 MDa was imposed. Before monitoring the diffusional motion, the particles were allowed to equilibrate for 5000 time steps. The remaining parameters were t = 10–9 s, rc = 2 nm, A/(6) = 103 µm2/s. |
RESULTS |
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= 1 ± 0.05 which indicates that finding anomalous subdiffusion with our setup is not an artifact of a distorted confocal volume (Hess and Webb, 2002; see also discussion in Weiss et al., 2003). Representative autocorrelation curves C() for dextrans of different molecular weight are shown in Fig. 2. The measurements in PBS also allowed us to determine the apparent hydrodynamic radius rH of the particles (see Methods). In the inset of Fig. 2 we show the increase of the radii for increasing molecular weight m (rH m0.4). In fact, the radii increase slower than anticipated for a simple random-coil polymer for which a description as a linear Gaussian chain yields rH m0.5 (Doi, 1996). This deviation is in agreement with previous reports (Cheng et al., 2002) and may be explained by the fact that dextrans become strongly branched polymers when their mass increases (Nordmeier, 1993).
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) are shown in Fig. 3. In strong contrast to the behavior in PBS, all dextrans showed subdiffusive motion in cytoplasm albeit with varying degrees of the anomality parameter . Moreover, the characteristic timescales D of the autocorrelation decays were increased with respect to the ones found for PBS which indicates an overall decrease of the diffusional mobility. Surprisingly, the determined degrees of anomality did not correlate linearly with the hydrodynamic sizes of the dextran particles (see Table 1). Rather, we observed a very strong subdiffusive motion for small dextrans (40 kDa) which relaxed for increasing mass (500 kDa) and then became stronger again (2 MDa). We next verified that the observed subdiffusion in cytoplasm was not a particular feature of dextran by monitoring the diffusion of a FITC-labeled IgG antibody (m 150 kDa) in cytoplasm. Having an apparent hydrodynamic radius rH 5.5 nm (cf. also Arrio-Dupont et al., 2000), we expected IgG to show a similar degree of subdiffusion as seen with 150 kDa dextran (rH 5 nm). In fact, we observed a stronger anomality ( 0.55, see also Fig. 3, inset), which may be explained by the fact that an IgG has a different shape than a 150 kDa dextran in solution.
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). In case iv the interior of the cell has undergone major changes due to the impending cell division, e.g., the microtubules form a spindle rather than an astral array. In agreement with our hypothesis, the subdiffusion persisted in all cases with similar values for (see summary in Table 2). This provides strong evidence that obstruction by higher-order structures is not the major cause of the observed subdiffusion. Rather, the observed subdiffusion is caused by molecular crowding.
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m
= 80 kDa. Bearing in mind that the used approach actually overestimates the fraction of small proteins due to the denaturing conditions in the gel (protein complexes are disrupted), we tested two distributions in the simulations which were inspired by the experimental distribution p(m) (see Fig. 4 a): i), a Poisson distribution with m = 350 kDa, and ii), a uniform distribution. In both cases we only considered proteins with masses up to 1 MDa and, for simplicity, assumed the proteins to be globular. In both simulation settings, we observed a size-dependent emergence of anomalous subdiffusion which also clearly depended on the fractional volume occupied by the globular proteins ("excluded volume"). In Fig. 4 b we show representative curves for the mean square displacement obtained for scenario i, i.e., a Poissonian distribution of molecular masses, at an excluded volume of 13%. Although small proteins were still diffusing more or less normally, the big particles clearly moved subdiffusively. This size-dependence is further highlighted in Fig. 4 c, where one can observe the decrease of the anomality parameter with increasing effective particle size. This result was only slightly altered in scenario ii, i.e., for a uniform size distribution. The decrease of with increasing radii persisted (Fig. 4 d) albeit occurring at bigger radii and at lower values for the excluded volume (7% instead of 13%). As both settings yielded the same gross features, we conclude that an excluded volume interaction (= molecular crowding) likely explains the subdiffusion observed in the cytoplasm of living cells. The successful simulations of course only represent the simplest possible configuration due to the use of globular particles. To quantitatively explain the experimentally observed -values, a more detailed approach may be necessary which includes, for example, the polymeric nature of the probe (see also Discussion).
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-values between in vitro and in vivo experiments using a particular probe. Consistent with our findings in vivo (the cytoplasm), we observed an increase of the diffusional time D and a concomitant decrease of the anomality parameter for the tested dextrans when the concentration C of unlabeled dextran (i.e., the crowding) in the solution was increased (Fig. 5). These experiments also confirmed the simulation results, i.e., the interaction via excluded volume can cause subdiffusion. In accordance with the results in living cells, we again observed that 40 kDa dextran appeared to be much more subdiffusive than its 500 kDa counterpart. We speculate that in both cases this may be caused by a partial reptational movement of the fairly short 40 kDa polymer whereas the more heavy dextrans may be more globular and are thus less prone to reptation (see also Discussion). Nevertheless, we conclude that the degree of anomalous diffusion () is a direct reflection of molecular crowding. By comparing in vivo measurements with those in vitro, one can therefore use the determined -values as a measure for molecular crowding.
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DISCUSSION |
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It is likely that the observed subdiffusion only persists for intermediate times and that normal diffusion is reencountered for asymptotically large times. For example, in our simulations we observed via the growth of the mean square displacement v(t) that even for a fairly low excluded volume subdiffusion transiently emerged on scales t < 1 µs and then collapsed back to normal diffusion. For increasing particle concentration this subdiffusive regime eventually extended beyond the 1 ms-scale (cf. Fig. 4). Similar phenomena are, for example, also found for obstructed diffusion with immobile obstacles near to the percolation threshold (Saxton, 2001) or for reptating polymers (Doi, 1996). Bearing this in mind, our results do not contradict but rather complement previous studies on cytoplasmic diffusion by means of photobleaching techniques (Seksek et al., 1997; Arrio-Dupont et al., 2000) which employ larger spatial and temporal scales than in FCS and therefore potentially miss the regime of subdiffusion.
In regards to the nature of the used probe, we observed that small dextran molecules can exhibit a much stronger anomalous subdiffusion than their more heavy counterparts (cf. Table 1 and Fig. 5). The most likely explanation for this phenomenon is a (partially) reptational movement of small dextrans. In the ideal case, reptation yields = 0.5 (Doi, 1996) whereas obstructed diffusion of globular particles typically yields a higher value for (see Introduction). For our case, we propose that small dextrans adopt a "snake-like" conformation whereas the more heavy dextrans are more globular and thus are rather subject to obstructed diffusion than reptation. This reasoning is supported by the fact that fructan, a close relative to dextran, was shown to behave like a random-coil polymer for masses 
whereas above 100 kDa it appeared more like a globule (Kitamura et al., 1994). This reasoning appears even more plausible when bearing in mind that the conformation of (bio)polymers can depend critically on the solvent and that dextrans show strong branching when their mass increases (Nordmeier, 1993). Of course, for reptational movement the simple picture used in the simulations becomes invalid and has to be replaced by a more elaborate polymer model in a heterogeneous environment. It will be interesting to study the crossover from reptation to obstructed diffusion in more detail (M. Weiss et al., unpublished results).
Despite the caveat that the observed subdiffusion may be a transient feature, it is still likely to play a major role in cytoplasmic processes. In our approach with FCS, we observed subdiffusion on a scale of 500 nm (the diameter of the confocal volume), a scale which is 100-fold bigger than the typical radius of a globular protein and almost corresponds to the typical size of an Escherichia coli bacterium. At least on this scale, anomalous diffusion can greatly modulate the interaction of proteins, e.g., in reaction networks (Berry, 2002; Saxton, 2002) and maybe in protein folding (Ellis, 2001; Hall and Minton, 2003).
Most importantly, the described emergence of subdiffusion provides a means to define a quantitative measure to what crowdedness actually means. In fact, the term "crowdedness" by its mere literal sense signals that the size and conformation of a test particle dictates if it feels an environment as being crowded. Being a water molecule, the cytoplasm does not appear to be any more crowded than any other solution. However, for a macromolecule, and even more for a polymer-like dextran, the cytoplasm with all its embedded proteins provides an obstacle-rich environment. We therefore propose that the degree of anomality can serve as a size- and conformation-dependent quantity to characterize the concentration/composition of a heterogeneous solution like the cytoplasm. In other words, by using a defined and standardized set of in vitro solutions (where the composition is varied), it should be feasible to use the degree of anomality as a quantitative measure to probe molecular crowding in vivo, be it in the cytoplasm, the nucleus, or other cellular or extracellular environments.
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ACKNOWLEDGEMENTS |
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The MEMPHYS-Center for Biomembrane Physics is supported by the Danish National Research Foundation. We also thank the Advanced Light Microscopy Facility (European Molecular Biology Laboratory, Heidelberg) and the SWEGENE Centre for Cellular Imaging at Göteborg University, Gothenburg, Sweden.
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FOOTNOTES |
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Submitted on April 9, 2004; accepted for publication August 16, 2004.
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G. Guigas and M. Weiss Sampling the Cell with Anomalous Diffusion The Discovery of Slowness Biophys. J., January 1, 2008; 94(1): 90 - 94. [Abstract] [Full Text] [PDF]
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B. D. Slaughter, J. W. Schwartz, and R. Li Mapping dynamic protein interactions in MAP kinase signaling using live-cell fluorescence fluctuation spectroscopy and imaging PNAS, December 18, 2007; 104(51): 20320 - 20325. [Abstract] [Full Text] [PDF]
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G. Guigas, C. Kalla, and M. Weiss Probing the Nanoscale Viscoelasticity of Intracellular Fluids in Living Cells Biophys. J., July 1, 2007; 93(1): 316 - 323. [Abstract] [Full Text] [PDF]
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M. J. Saxton A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model Biophys. J., February 15, 2007; 92(4): 1178 - 1191. [Abstract] [Full Text] [PDF]
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H. Sanabria, Y. Kubota, and M. N. Waxham Multiple Diffusion Mechanisms Due to Nanostructuring in Crowded Environments Biophys. J., January 1, 2007; 92(1): 313 - 322. [Abstract] [Full Text] [PDF]
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D. Weihs, T. G. Mason, and M. A. Teitell Bio-Microrheology: A Frontier in Microrheology Biophys. J., December 1, 2006; 91(11): 4296 - 4305. [Abstract] [Full Text] [PDF]
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C. Pack, K. Saito, M. Tamura, and M. Kinjo Microenvironment and Effect of Energy Depletion in the Nucleus Analyzed by Mobility of Multiple Oligomeric EGFPs Biophys. J., November 15, 2006; 91(10): 3921 - 3936. [Abstract] [Full Text] [PDF]
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D. Bhattacharya, A. Mazumder, S. A. Miriam, and G. V. Shivashankar EGFP-Tagged Core and Linker Histones Diffuse via Distinct Mechanisms within Living Cells Biophys. J., September 15, 2006; 91(6): 2326 - 2336. [Abstract] [Full Text] [PDF]
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A. Delon, Y. Usson, J. Derouard, T. Biben, and C. Souchier Continuous Photobleaching in Vesicles and Living Cells: A Measure of Diffusion and Compartmentation Biophys. J., April 1, 2006; 90(7): 2548 - 2562. [Abstract] [Full Text] [PDF]
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