INTRODUCTION

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Proteins in cell membranes typically diffuse much more slowly than do proteins in artificial lipid bilayers. Although the simplest hydrodynamic theory predicts that lateral diffusion of proteins in cell membranes should be roughly half as fast as that of lipids (Saffman and Delbruck, 1975), the experimentally measured ratio is often one to three orders of magnitude (Jacobson et al., 1987). Both long-range and short-range protein diffusion is slower than expected from interaction with lipids.

Much has been learned recently about retardation of long-range diffusion. It is clear that in many cells, long-range diffusion is restricted by confinement zones of 250-1500-nm diameter. In such zones, proteins reside from ~3-35 s before they move to the next confinement zone (Saxton and Jacobson, 1997). This has been attributed to encounters with a "membrane skeleton fence" in fibroblasts (Sako and Kusumi, 1995). Within each confinement zone, diffusion is relatively fast, but long-range diffusion is slowed by barriers between compartments.

Even "fast" diffusion of proteins within such compartments, however, is much slower than in pure lipid bilayers. Therefore, factors not readily apparent by simple single particle tracking (SPT) measurements operate to retard diffusion on the micro scale, independent of the barriers that have been described. This freer, but still retarded, diffusion within compartments is characterized by D_micro, the diffusion coefficient within a compartment, determined from the initial slope of the mean-square displacement plot versus the time interval (Sako and Kusumi, 1995).

The three basic models of retardation of diffusion, as we envision them, are depicted in Fig. 1. The first involves obstruction by membrane microcorrals (Fig. 1 A). This model proposes that within confinement zones there exist a similar set of weaker barriers, too small to be detected directly on the time scale of SPT. These would behave much like the larger confinement zones, slowing diffusion over distances greater than the span of the microcorral. These microcorrals would have to be <~0.1 µm across, however, since they are not readily detectable by examination of SPT particle tracks. Direct evidence for interactions of membrane proteins with corrals of this size is lacking. It has been suggested, however, that obstruction by such barriers may account for the gap between hydrodynamic theory and experimental measurement in some systems (Bussell et al., 1995a; Dodd et al., 1995).

View larger version (42K): [in this window] [in a new window]   FIGURE 1   Models of retardation of diffusion on the micro scale. (A) The microcorral model. This model proposes that within confinement zones there exists a similar set of weaker barriers, too small to be detected directly on the time scale of SPT. These barriers retard membrane protein diffusion over distances greater than the span of the microcorral. Because of their small size (<~0.1 µm across), these microcorrals would not be detectable by examining SPT particle tracks. (B) The transient binding model. This model predicts that the diffusion of cell surface proteins is impeded by interactions between the mobile proteins and slowly moving or immobile structures in the cytoplasm, e.g., cytoplasmic proteins bound to the cytoskeleton (cf Zhang et al., 1991). The example illustrated depicts an indirect interaction with the cytoskeleton via a single cytoplasmic protein; a variety of other indirect links would have the same effect. If the lifetime of the bound state were short compared to the measurement time, the diffusion coefficient would be reduced by the fraction of the time it spent in the bound state (Elson and Reidler, 1979). (C) A viscous model. This illustration depicts protein crowding as the source of high apparent viscosity. Alternatively, high apparent viscosity might be due to other sources of drag, such as viscous interactions with intra- or extracellular structures (Zhang et al., 1991), excluded area effects (Saxton, 1990), or hydrodynamic effects of immobile proteins (Bussell et al., 1995b).

While identification of the specific proteins comprising these barriers to diffusion is beyond the scope of this study, we suggest that a candidate protein would be actin itself. Electron micrographs of the lamella of the fish epidermal keratocyte show that the actin cytoskeleton underlying the plasma membrane in the lamella forms a tight meshwork, the size of which is consistent with microcorrals (Svitkina et al., 1997). Such an actin network configuration is also typical of many fibroblastic cells.

The second model involves rapid and repetitive transient binding to and release from slowly moving or immobile structures, on a time scale so fast that individual binding events would not be discernible by SPT measurements (Fig. 1 B). Although this is not a new idea, recent support for this model comes from experiments with band 3, the major membrane protein of the erythrocyte (Golan et al., 1996). In that system, both lateral and rotational mobility of band 3 was higher in erythrocytes deficient in the protein band 4.2 than in normal cells. Since band 4.2 binds both band 3 and elements of the cortical cytoskeleton, a plausible explanation for this retardation of diffusion would be that transient binding to the cytoskeleton, via band 4.2, retards both rotational and lateral motion of band 3. As pointed out by Bussell et al. (1995a), not all of the retardation of band 3 can be explained by current hydrodynamic theory. The typical nucleated cell, however, is very different from the erythrocyte with a much less structured cytoskeleton underlying the plasma membrane. Strong evidence for or against transient binding to explain retardation of diffusion in nucleated cells is, to our knowledge, currently lacking.

Finally, simple viscous resistance to membrane protein motion might slow diffusion on the micro scale. This viscous resistance, however, would have to be much greater than that expected from the lipids of the membrane bilayer itself. One likely source of this increased apparent viscosity is the effect of other membrane proteins, especially immobile ones (Fig. 1 C). Relatively few immobile proteins can have profound effects on the effective viscosity of the plasma membrane. In this model, the diffusion of a "tracer" protein is slowed by the effect of other proteins embedded in the same lipid bilayer. This represents the current state of the hydrodynamic model (discussed in more detail in the Discussion section) (Bussell et al., 1995a, c; Dodd et al., 1995). It is important to note, however, that our study does not distinguish among possible sources of increased viscosity. Therefore, this model is meant to encompass increased viscosity due to any source, including protein crowding (excluded area effects) (Saxton, 1990), hydrodynamic effects of mobile or immobile proteins (Bussell et al., 1995a, c; Dodd et al., 1995), viscous interactions with the cytoplasm or with the glycocalyx coating the cell's outer surface (Zhang et al., 1991), or a combination of all of these.

While all three models predict similar behavior for individual membrane proteins, they differ in their predictions for the rate of lateral diffusion of membrane proteins clustered in aggregates of various sizes. In particular, the first two, which involve direct interaction with the cytoskeleton, predict a strong dependence on aggregate size, while the last, which involves only viscous interactions, does not. We formed membrane protein aggregates by coating beads (ranging in diameter from 40 to 550 nm) with con A and allowing them to attach to membrane proteins on the cell surface. Theory predicts that the region of contact between a bead and the cell surface should increase with the size of the bead (see Discussion section and Appendix). The number of glycoproteins bound should also increase with the contact area, a consideration important for evaluating the transient binding and model. By using this approach we have been able to discriminate among models.

To measure diffusion rates, we used FRAP for individual proteins (Axelrod et al., 1976), and used SPT (Sheetz et al., 1989; Qian et al., 1991; de Brabander et al., 1991) for bead-induced aggregates. SPT uses small ligand-coated gold or latex beads to tag membrane proteins. Larger beads form aggregates by accumulating a number of protein molecules proportional to the area of contact between the bead and the cell membrane. These are then tracked with high resolution using video-enhanced DIC microscopy. Random and directed components of motion can be separated unambiguously (Sheetz et al., 1989; for a detailed discussion of theory, see Qian et al., 1991) and movement of the cell can be subtracted (Kucik et al., 1990). We used con A, a lectin that binds many species of membrane glycoproteins, to coat the beads. Thus, the behavior of the beads should reflect a general property of membrane proteins, rather than interactions specific to a particular membrane protein.

We made these measurements on fish epidermal keratocytes (FEKs) because of their favorable optical properties and the relative lack of surface features, such as ruffles, which might influence diffusion (Kucik et al., 1990). Also, in many cell types, large con A-coated beads often induce nondiffusional behavior, e.g., rearward transport, perhaps due to cross-linking of certain receptors (unpublished observations). In FEKs, however, although a few beads undergo directed transport, the vast majority (>95%) of con A-coated beads of all sizes display only diffusional behavior for several minutes (Kucik et al., 1991).

We measured the diffusion coefficients of membrane proteins by both FRAP and SPT in the same system, FEKs, with the same membrane protein ligand, con A, always at room temperature (a physiologic temperature for goldfish). Because we could not directly measure the area of contact between the beads and the cell membrane, we used a model calculation to predict how the size of the aggregate should vary with the bead diameter (see Appendix). As previously observed on FEKs and other cell types, the measured rates of diffusion (D_micro) for the smallest beads were orders of magnitude slower than theoretically expected for ideal diffusion in a pure lipid bilayer, but well within the range of actual FRAP measurements of diffusion coefficients of individual membrane proteins on a variety of living cells (Jacobson et al., 1987). This suggests that the same factors that slow protein movement on more commonly studied cells also operate in the FEK cell system, making it a valid system to study.

The dependence of diffusion on aggregate size was found to be very weak. Equally striking, diffusion coefficients of beads measured by SPT do not differ substantially from those of individual glycoproteins labeled with fluorescein-labeled succinyl con A and measured by FRAP. These observations are inconsistent with "transient binding" and "microcorral" models, but favor a hydrodynamic model, i.e., that the membrane proteins move as if embedded in a very viscous two-dimensional fluid (discussed below).

	MATERIALS AND METHODS

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Cells

Scales were removed from goldfish (Carassius auratus), and placed on acid washed glass coverslips in the presence of bovine serum (Kolega, 1986). FEK cells that crawled onto the glass were then cultured in fish Ringer's supplemented with Amphibian medium obtained from Gibco (Life Technologies, Inc., Gaithersburg, MD) as described in Cooper and Schliwa (1986). The coverslips were then transferred to a stage medium of fish Ringer's for microscopy.

Single particle tracking

Latex beads (Polysciences, Inc. Warrington, PA) and gold beads (Janssen Pharmaceuticals, Piscataway, NJ) were coated with con A by adsorption as described earlier (Sheetz et al., 1989). These were added to the cells in a Ringer's solution stage medium, their motions were observed by video-enhanced differential interference contrast microscopy, and images were recorded on sVHS videotape for later analysis. Bead positions in each frame were determined by computer by the method of Gelles et al. (1988). From these position measurements diffusion coefficients were determined as described in Results and Discussion.

Con A-coated colloidal gold particles (40-nm diameter) and latex beads (190- and 550-nm diameter) bound to the cell surface via membrane glycoproteins. Beads were allowed to bind passively (40-nm gold) or were placed on the cell with laser tweezers. This binding was specific: it could be blocked by incubating the con A-coated particles with glycoproteins before an experiment (data not shown). Whether the particle was diffusing freely in the membrane or was anchored to the cytoskeleton was determined unambiguously from the diffusion coefficient of each bead as previously described (Sheetz et al., 1989; Kucik et al., 1989). Briefly, diffusion coefficients of diffusing beads are orders of magnitude greater than those of beads anchored via membrane proteins to the cytoskeleton (Sheetz et al., 1989; Kucik et al., 1990). Nearly all beads (>95%) diffused randomly, except for those very near the edge of the cell or a few that underwent cytoskeleton-driven rearward transport. The motion of these atypical beads was not analyzed.

The SPT measurements were performed on both locomoting and stationary cells. On locomoting cells diffusing particles move passively along with the cell, as described earlier (Kucik et al., 1990): in the frame of reference of the cell the particle motion is completely random. Brownian motion of the particle was analyzed independently of the directed motion component (Qian et al., 1991); see also Results section and Fig. 2.

View larger version (27K): [in this window] [in a new window]   FIGURE 2   Examples of data produced by the SPT method and their analysis. Panels (A) and (B) are examples of particle tracks of 40-nm gold and 550-nm latex beads, respectively. Particle tracks yielding diffusion coefficients near the mean for each group were chosen for this example. In this case, each particle track is 363 points in length, each point representing a particle position measurement. Measurements were taken every 1/30 s, i.e., video frame rate. Bar = 1 µm. Mean length of particle tracks analyzed was 608 points = 20 s. Panels (C) and (D) are plots of msd of the particles in (A) and (B), respectively. For a particle diffusing in two dimensions, msd = 4Dt, where D is the diffusion coefficient. The displacement d of a particle undergoing directed motion at constant velocity V is described by d = Vt and msd = (Vt)2. Hence the random and directed components of particle motion can be extracted by fitting the msd to the equation msd = 4Dt + (Vt)2, and the diffusion coefficient can be determined [for a complete discussion of this analysis, see Qian et al. (1991)].

Fluorescence photobleaching

FITC-s-con A was obtained from Sigma Chemical Co., St. Louis, MO. Fluorescence photobleaching measurements of labeled proteins in keratocyte membranes labeled with FITC-s-con A were carried out as previously described (Dubinsky et al., 1989).

Because we are assuming purely random motion of fluorophores (Axelrod et al., 1976), directed motion of unbleached fluorophores into the bleached spot can yield an artefactually elevated diffusion coefficient (D). Therefore, we were careful to eliminate any component of directed motion of the fluorophore. Apparent directed motion can result from drift of the microscope stage, cell migration, or systematic transport of cell surface features or particles. Systematic drift of the microscope stage relative to the laser beam was ruled out by occasionally bleaching a spot in the FITC-s-con A adsorbed to the glass coverslip in regions devoid of cells. The immobilized fluorophores indicated that drift of the microscope stage relative to the laser beam was undetectable in our measurements. Distortion of the FRAP measurements by active motions of normal cells was suppressed by treating the cells with 10 mM CoCl₂ or cytochalasin D (2 µg/ml). Cobalt ion frequently paralyzes the cells without altering their morphology or their actin cytoskeleton (Cooper and Schliwa, 1986). Although cytochalasin D caused most cells to retract their lamellae, there were enough cells which remained both spread and immobile so that diffusion measurements could be carried out. All diffusion measurements were made on the lamellar portion of the cell.

It has become clear in recent years that many membrane proteins are neither randomly diffusing nor immobile, but are undergoing systematic transport. Contributions of systematic fluorophore transport to fluorescence recovery can be assessed by varying the magnification of the objective lens, and therefore the size of the bleached spot. For simple diffusion the characteristic time for fluorescence recovery, _D, varies as the square of the radius of the photobleached spot, w, i.e., _D = w²/4D, where D is the diffusion coefficient (Axelrod et al., 1976). Contributions from transport mechanisms other than random diffusion cause the characteristic recovery time to vary from a simple dependence on w². Although most of our measurements were made with a 100× objective (w = 2.1 µm), the 40× (w = 0.84 µm) was occasionally used to rule out nonrandom motion of fluorophores. Comparison of measurements made with the two objectives often detected nonrandom motion in unparalyzed cells, usually due to movement of the cell itself. After the cells were paralyzed by CoCl₂ or cytochalasin D, however, systematic drift was never detected by this method. In addition, each cell was carefully observed before and after FRAP measurements to control for motion of the cell or its surface features.

Electron microscopy

Shortly after the addition of 0.5-µm con A beads, the cells were fixed according to a procedure described previously (Martenson et al., 1993). Fixative was added while the sample was being viewed by video-enhanced DIC microscopy to determine that cell morphology was not altered by fixative. After fixation and embedding, the glass microscope slide was removed and the specimen embedded in epon was examined to identify the regions of interest. Those regions were cut out and re-embedded in epon for sectioning with an orientation such that ultrathin sections were cut perpendicular to the original glass surface with a diamond knife. Sections were viewed on a Phillips 301 microscope.

	RESULTS AND DISCUSSION

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Sixty-six beads were tracked by SPT on 12 different days. In each experiment, beads of a particular size were added to a cell chamber, and motion was observed with video-enhanced DIC and recorded onto sVHS videotape. Fig. 2, A and B are examples of particle tracks generated by a 40-nm con A-coated gold bead and a 550-nm con A-coated latex bead. Such particle tracks result from frame-by-frame analysis of video sequences, with particle positions accurately determined by the method of Gelles et al. (1988). Plots of mean square displacement versus time were generated from these particle tracks (Fig. 2, C and D). Under our experimental conditions, >95% of beads of all sizes diffused freely. Those that did not usually were only transiently immobilized. Particle tracks were analyzed, and diffusion coefficients (D) were calculated for those whose motion was random (not directed or restricted in the frame of reference of the cell).

While movement of the cell during the time course of the measurement contributed a directed component to the particle track (in the frame of reference of the lab), SPT measurements permit the separation of the random and systematic contributions to the motion of an individual bead, as previously explained (Sheetz et al., 1989; Qian et al., 1991). Briefly, purely random motion results in a linear increase in msd with elapsed time, t, i.e.,

(1)

For a constant velocity component of directed motion (such as that contributed by movement of the cell), d = Vt and

(2)

where d = displacement and V = velocity. Thus, for a diffusing particle on the surface of a cell moving at constant velocity,

(3)

By fitting this quadratic equation to the data, the random diffusion component of motion can be separated from the directed movement of the cell, and a diffusion coefficient can be determined (Qian et al., 1991; Kucik et al., 1989). Fig. 3 summarizes the results of this analysis of 66 diffusing particles. The mean D values of the smallest and the largest particles differ by less than a factor of 2. 

View larger version (21K): [in this window] [in a new window]   FIGURE 3   Distribution of diffusion coefficients obtained from single particle tracking (SPT) data. While the mean D of the 550-nm beads is significantly different from that of the 40-nm beads and of the FITC-s-con A-labeled membrane proteins (see Table 1), the difference is small (less than a factor of 2) This small difference is inconsistent with transient binding and corral models and supports a hydrodynamic model of retardation of diffusion (see Discussion).

Diffusion coefficients obtained by SPT were compared to those obtained by photobleaching FITC s-con A-labeled membrane proteins. Like con A, s-con A labels a variety of membrane proteins, but is thought to cause less cross-linking due to its lower valency. Thus we could examine the behavior of individual membrane proteins or, at the most, small oligomers. Since FEKs are rapidly motile cells and fluorescence photobleaching is not as well suited as SPT to separating random motion of membrane proteins from directed motion of the cell, it was necessary to paralyze the cells to prevent movement of the bleached spot on the time scale of the measurements. We did this with cytochalasin D (2 µg/ml), which disrupts actin filaments (sometimes causing cells to round up; measurements were made only on cells which retained flat lamellae), and in separate experiments with CoCl₂, which paralyzes FEKs without greatly altering their morphology or disrupting the actin cytoskeleton (Cooper and Schliwa, 1986).

A total of 14 measurements were made on cytochalasin D-treated cells, and 50 measurements on CoCl₂-treated cells. The diffusion coefficient of s-con A on FEK calculated from these measurements was 8.6 ± 4.6 × 10¹⁰ cm²/s for CoCl₂-treated cells, and 7.2 ± 2.8 × 10¹⁰ cm²/s for cytochalasin D-treated cells (Fig. 4). The data obtained with the two methods of paralyzing cell movement are in good agreement with each other. These values are also well within the range of diffusion coefficients measured FRAP for many different membrane proteins on a variety of other cells (Jacobson et al., 1987). Data from all FRAP and SPT measurements are summarized in Table 1.

View larger version (30K): [in this window] [in a new window]   FIGURE 4   Distribution of diffusion coefficients obtained from fluorescence photobleaching data. Membrane proteins were labeled with FITC-s-con A and their diffusion coefficients were measured by bleaching a spot and monitoring the time course of recovery of fluorescence. Since FEKs are highly motile cells, all cells were paralyzed for these experiments so that the bleached spot would not move during the measurement due to cell movement. Both cytochalasin D and cobalt chloride (cf. Cooper and Schliwa, 1986; also see text) were used to paralyze cells. The magnitudes and distribution of the diffusion coefficients agree well with typical values for membrane proteins on many types of cells (Jacobson et al., 1987).

Our major experimental result, therefore, is that protein aggregates induced by con A-coated beads, some of which are quite large, diffuse almost as fast as single membrane proteins. A trivial explanation for these results would be that a con A-coated bead binds to only one or a few membrane glycoproteins independent of its size. This, however, is unlikely. Measurements of protein adsorption on colloidal gold particles predict that the number of proteins bound depends on the size of the protein and the size of the particle in a predictable manner (De Roe et al., 1987). According to this analysis, our smallest particles, 40 nm, should bind ~240 con A molecules. Particles carrying so many multivalent ligands should form many links to membrane proteins, providing that the membrane proteins are relatively abundant. To establish directly that our largest beads were forming protein aggregates, we examined the contact area between the 500-nm beads and the cell surface by electron microscopy. As shown in Fig. 5, the large beads form extensive areas of contact with the membrane.

View larger version (152K): [in this window] [in a new window]   FIGURE 5   Electron microscopy illustrates extensive contact between con A-coated 0.5-µm latex beads and the plasma membrane. While there is a range of contact areas, illustrated by panels (A) and (B), the beads were never seen to be attached at a single point.

The size of a membrane protein aggregate formed by such a bead will depend on the contact area of the bead with the membrane. Assuming a constant density of con A molecules per unit area of the microspheres (De Roe et al., 1987), larger beads, with larger contact areas, will form larger membrane protein aggregates. Although we do not have a direct measure of the change in contact area with bead size in this system, it is possible to analyze the functional dependence of contact area on bead diameter. This analysis yields a dependence of contact area on R², where R is the radius of the bead. Hence, n, the number of con A molecules linked to a bead, should also vary as R². A detailed discussion of this is provided in the Appendix.

The predicted behavior of such bead-induced protein aggregates in random motion in the plane of the membrane is different for the three models of retardation of diffusion that we considered. We interpret these results in terms of various models for retardation of diffusion, as follows.

	MODELS FOR MEMBRANE GLYCOPROTEIN DIFFUSION

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The microcorral model

According to the microcorral hypothesis, diffusion in the cell membrane is limited by barriers which form restrictive enclosures, or "microcorrals" (Sheetz, 1983). This term is used to distinguish small divisions of the membrane, below the resolution of SPT on a 33-ms time scale, from the larger "confinement zones" (0.25-1.5 µm diam.) which are readily visible in particle tracks. These barriers might be composed of submembrane filaments that are constantly undergoing rearrangement. In the absence of hydrodynamic effects caused by these immobile structures (Bussell et al., 1995a; Dodd et al., 1995), within a microcorral a membrane protein could move rapidly, at a rate limited only by lipid viscosity. While not directly observable by conventional means, the rate of diffusion within microcorrals could then be as high as that measured for proteins in artificial lipid bilayers at 25°C, ~2 × 10⁸ cm²/s (Tank et al., 1982). To diffuse over distances greater than the dimensions of the microcorral, however, a barrier would have to be removed to allow the protein to pass into the adjacent microcorral. Although diffusion within microcorrals might be slower than in reconstituted membranes because of hydrodynamic effects from immobile proteins, breakdown of these barriers might still be the dominant influence on lateral mobility of membrane proteins. Therefore, we consider the possibility that very small corrals, which would not be directly observable, i.e., with dimensions on the order of 0.1 µm, might account for the observed slow diffusion (Sheetz, 1983, 1995; Kusumi et al., 1993; Saxton, 1994, 1995). This model would explain retardation of individual proteins, but would predict much more retarded motion for large protein aggregates. In the present studies, aggregates formed by the larger beads would be expected to span across boundaries of several microcorrals. Assuming that these barriers break down independently with a probability P_b, the probability that n barriers would simultaneously disappear is (P_b)ⁿ. The number, n, of submembrane barriers interacting with the bead at any moment should be proportional to the area of contact between bead and cell membrane. The rate-limiting step for diffusion would, therefore, depend exponentially on n, which should in turn depend on R², i.e., (P_b)ⁿ ~ (P_b)^R2.

The value of R² increases 189-fold between the 40- and 550-nm beads. If P_b is not too close to 1, this should be sufficient to rule out the microcorral hypothesis. For example, if P_b = 0.98 (a transient barrier that exists only 2% of the time), (P_b)^R2 = 0.02. A 50-fold change in diffusion rate would have been easily detectable in our measurements, but was not seen. Any value of P_b < 0.98 (representing a more substantial barrier) predicts an even stronger dependence of D on aggregate size. Hence the microcorral model is ruled out by our experimental measurements.

Transient binding models

A transient binding model predicts that the diffusion of cell surface proteins is impeded by interactions between the mobile proteins and slowly moving or immobile structures in the cytoplasm, e.g., cytoplasmic proteins bound to the cytoskeleton (cf. Zhang et al., 1991). These interactions could be direct or indirect. An equivalent situation would be transient binding to external structures, e.g., glycosyl moieties on the surface of the cell, or the extracellular matrix. If the lifetime of the bound state were long compared to the time required for measuring diffusion rate, the bound protein would appear immobile. If, however, the lifetime of the bound state were short compared to the measurement time, the protein molecule would bind and detach many times during the measurement. It would, therefore, appear mobile, with its diffusion coefficient reduced by the fraction of the time it spent in the bound state (Elson and Reidler, 1979).

A realistic analysis would allow for independent formation and breakage of bonds within the contact region. If there is a probability P that each membrane protein will be transiently immobilized at any instant, and a probability Q that it will be diffusing freely, then the probability that at least one of a group of n proteins on the surface of a bead will be immobilized at any time is 1  Qⁿ. Although the molecular mechanism is different from that of the microcorral model, i.e., binding versus steric confinement, the consequences for protein aggregates are similar. Assuming a uniform density of con A on the beads, an exponential dependence on n again results in an exponential dependence on R². If even one of the proteins attached to the bead is immobilized at any given time, the bead will be immobile. Thus, as the number of con A receptors attached to the bead increases, the probability that the bead will be immobile at any time rapidly approaches 1. A strong dependence of D on bead size is thus predicted, so a transient binding model is inconsistent with our data.

Viscous models

Since we do not know the extent of permeation of lipid molecules into the bead-induced protein aggregates, it is necessary to consider two classes of hydrodynamic models. The consequences for retardation of protein diffusion are different, depending on whether the aggregates behave as impermeable cylinders or are free-draining. We cannot directly measure the density of membrane proteins bound to our beads. We can, however, infer from our data that these proteins are diffusing as unit aggregates, as follows.

Free-draining aggregate diffusion

Unit aggregate diffusion

(4)

5