(Un)Confined Diffusion of CD59 in the Plasma Membrane Determined by High-Resolution Single Molecule Microscopy

INTRODUCTION

The cellular plasma membrane has shifted into the spotlight of cell biologists because it represents the major regulatory platform for the initiation of early signaling events. Signal transmission is generally enabled by a sequence of tightly regulated protein interactions, which trigger intracellular second messenger release. A comprehensive model of early signaling events therefore requires understanding the physical principles that mediate and affect interactions between proteins in the plasma membrane. Currently, the concepts are as indecisive as can be: a majority of researchers has accepted the view that small lipid domains—so-called lipid rafts—should segregate membrane proteins into two exclusive fractions: a raft fraction comprising basically signaling molecules, and a nonraft fraction including, e.g., the majority of transmembrane proteins (1–3). However, a growing number of scientists have articulated their skepticism (4,5) based on new experimental insights into potential artifacts associated with the original supporting studies (6).

An additional aspect has been recently introduced into the field, when the mobility of various membrane constituents has been measured in unprecedented detail: although there was evidence for years that transmembrane proteins interact with the membrane skeleton underlying the cytosolic leaflet of the cellular plasma membrane (7), single particle tracking at 25 μs time resolution revealed that even the diffusion of gold-labeled phospholipids or lipid-anchored proteins in the exoplasmic leaflet was affected by the membrane skeleton (8,9); transient confinement to periodic corrals with a size ranging from 32 to 230 nm for different cell types has been detected (9). This model of hop diffusion appeared attractive because it provides an elegant explanation for the observed increase in the apparent viscosity of the plasma membrane compared to artificial lipid bilayers (10–12). Because confinement of probes without direct contact to the confining elements was difficult to interpret, lipid rafts were suggested as the most promising transmitting elements. The residence time of a lipid within a corral would then correlate with the lifetime of the metastable raft itself, which the probe lipid is part of; based on these studies, lifetimes of only a millisecond or less are frequently attributed to rafts (13). However, these studies are also prone to errors: it has been found that gold labeling affects the motion of the diffusing probe in studies on model membranes (14) and cell membranes (15). The reason might be the size of the gold label that exceeds the size of the probe molecule severalfold as well as potential cross-linking of probe molecules by the gold particle that is coated with an antiprobe reagent.

Here, we investigated the mobility of CD59—a GPI-anchored protein—in the plasma membrane of living T24 (ECV) cells using single molecule microscopy. By employing minimum invasive labeling via fluorescent Fab fragments the recorded single molecule trajectories closely reflect the movement of an unlabeled protein. To sense confinement zones at the reported size of 120 nm (9), we dramatically improved the resolution in space and time of state of the art single molecule imaging devices down to 22 nm and 1 ms, respectively. For appropriate data interpretation, we derived an analytical approximation describing the time dependence of mean square displacements for diffusing molecules impeded by an infinite array of partially permeable barriers.

THEORY

Analytical approximation for hop diffusion

To quantitatively characterize hop diffusion, we derived an analytical approximation for the according mean square displacement (see Table 1 for list of abbreviations). Diffusion within impermeable square corrals of size L can be described by the mean square displacement with D_micro the short-range diffusion constant of the molecule within the confined area (); f has been calculated by Powles et al., yielding (16). Its asymptotic behavior for long time-lags is given by .

TABLE 1.

Summary of abbreviations used in the Theory section

	Apparent fraction of fully confined motion in hop diffusion
CO	Confinement offset, corrected for localization errors
COcorr	Confinement offset, corrected for localization precision and illumination time effects
COreal	Measured confinement offset
Dmacro	Macroscopic diffusion constant
Dmicro	Microscopic diffusion constant
Δ	Effect of localization precision on MSD
f	MSD for totally confined diffusion according to Powles et al. (16)
L	Size of square corral
MSD	Mean square displacement
MSDreal	Measured mean square displacement
σx, σy, σxy	Localization precision in x- and y-direction (assumed to be equal)
σCO	Error of CO
	Error of COcorr
	Error of COreal
	Error of Δ
	Error of Dmacro
tdel	Time delay
till	Illumination time
	Normalized illumination time
tlag	Time-lag
τ	Residence time of a confined molecule
τmicro	Residence time of a freely diffusing molecule in a corral of size L
	Confinement strength; normalized residence time of a confined molecule

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If the corrals have a nonzero permeability, a molecule can escape and hop from corral to corral. For long observation times, the molecule appears to diffuse freely with a macroscopic diffusion constant , where describes the residence time within a corral. MSD increases with increasing time-lag and converges to

(1)

In this scenario, the confinement offset CO contains the information about the size of the confinement zones. Considering scaling invariance, can be regarded as a sole function of the dimensionless variable . sets the residence time of a confined molecule, , in relation to the time a freely diffusing molecule would stay in the same region; per definition . As a rule of thumb, can be interpreted as the confinement strength. In general, partially permeable corrals will lead to a decrease in the offset, which will vanish when the barriers are totally removed.

In the following we derive an analytical approximation for MSD_hop (supplemental Fig. 1, Supplementary Material). For this, we estimate the distance a molecule moves between two independent observations separated by the time-lag t_lag, characterized by the start position and the end position . Let us begin with an impermeable corral, and introduce weak permeability. The physical origin may be a single gate in the corral boundary. Whenever the molecule hits the gate, it will escape the corral with probability 1. Let us further characterize the trajectory of the molecule by time-averaging its positions within each visited corral, yielding in particular the average starting position and end position . In this scenario of quasiimpermeable boundaries, the time-averaged positions lie in the corral center. In the limit , the time-dependent component of the mean square displacement is given by the movement of the average position yielding . The actual position of the molecule within the domain at the time point of the observation will deviate from by ; according to Powles et al. (16), the distance is given by . Noting that this additional distance has to be considered for both the start and end position of the molecule, we estimate .

FIGURE 1.

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When we increase the permeability of the corral boundary, a significant fraction of the molecular motion will mediate the transition to the adjacent corral. This fraction is characterized by free diffusion with diffusion constant and therefore does not contribute to the confinement offset. The remaining fraction of the trajectory will experience no transitions, and is therefore described by totally confined diffusion. We account for the contributions of the two fractions by the modification .

Moreover, due to fractions of free diffusion in the trajectory the apparent extension of the observable corrals is reduced. We can estimate the area portion of the free diffusion subfraction by . In the remaining area the random walker experiences full confinement. The total movement of a random walker in partially permeable corrals is therefore well approximated by .

This equation provides an analytical approximation for hop diffusion in square corrals; its agreement with Monte Carlo simulations for different values of is shown in Fig. 1 A. The asymptotic behavior for short time-lags approximates unbounded motion. For long time-lags, we find , which converges to for (Fig. 1 B). It has to be noted that for finite the shape of the fully confined subsections may deviate from a perfect square; however, the deviations are quantitatively negligible in the context of this study.

In addition, movements of the molecule during the illumination time affect the results. For free Brownian motion with diffusion constant D_macro, a negative contribution to the offset has to be taken into account (17). For hop diffusion, a qualitative estimate (18) and a quantitative approximation (19) have been given in the literature, which we extend here by a more general analytical expression. The following heuristic argument assumes that even for impermeable barriers a confined molecule shows free diffusion, although only for a short period of time , i.e., with the probability , no effects due to confined molecular movements during the illumination time are observable. On the contrary, with a probability 1 − p the molecular motion collapses to the center of the confinement region, which reduces the respective mean square displacement to zero. This approach provides a valid approximation in the parameter range of our experiments, as tested by Monte Carlo simulations (Fig. 1 C). Considering all above contributions, a total mean square displacement for hop diffusion

(2)

is derived.

Assuming rapid free Brownian motion inside the corrals (D_micro ∼ 10 μm²/s), recording of the full time course of MSD represents a challenging task, as the initial rising phase would occur at . Still, the asymptotic behavior of MSD with contains information about confined diffusion due to an additional offset CO (Eq. 1 and Fig. 1):

(3)

In an experimental realization, the nonzero localization precision has to be further taken into account by

(4)

with , denoting localization precision, i.e., the standard deviation in a data set of positions from consecutive images of a single immobile molecule.

Error analysis

For error propagation analysis, we accounted for contributions due to errors in determination of the confinement offset , , and of the macroscopic diffusion constant D_macro, (for illustration, see Supplemental Fig. 2, Supplementary Material). First, the confinement offset was corrected for illumination time effects, . Using Gaussian error propagation analysis according to the expected error is given by

FIGURE 2.

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For Fig. 10, we determined the parameter duplet , which yields consistency with the measured values , , in a one-σ-range: . Note that both CO_corr and are functions of L and .

FIGURE 10.

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MATERIALS AND METHODS

RESULTS

The fact that single molecules can be localized with subpixel precision beyond the diffraction limits of light microscopy has enabled a manyfold of applications, in which the structure of biomolecules or biomolecular assemblies was studied on the submicrometer length scale (21–24). A key demand for confinement analysis is precise knowledge of the intrinsic errors in position determination, . We therefore calibrated our system by measuring the localization precision for all experimental conditions used in the further experiments. Cells were labeled with a fluorescent antibody to CD59, fixed, and single molecule trajectories were recorded and analyzed (Fig. 4). Within experimental errors, MSD was found to be constant when plotted as a function of the time-lag. From the offset of the curves, was calculated according to Eq. 4. We determined the localization precision under various illumination conditions, and plotted as a function of the average number of counts detected from a single molecule (Fig. 5). The expected proportionality (25,26) was confirmed quantitatively and used to correct all further experimental results according to .

Fig. 6 shows typical raw data for CD59 molecules moving in the plasma membrane of living T24 (ECV) cells. Before the experiment, cells were stained with a low concentration of Alexa647-labeled Fab-fragments to CD59; thereby, only a minority fraction of cellular CD59 was observable. In general, surface densities of ∼1 fluorescent CD59 molecule per 10 μm² were adjusted throughout our studies. An image sequence was recorded with t_delay = 0.4 ms and t_ill = 0.65 ms, and the position of individual molecules was determined on each image by fitting with a two-dimensional Gaussian profile. At the highest magnification shown, the trajectory of a single CD59 molecule is plotted as overlay (red points); due to the high temporal resolution, the total recorded molecular motion remains within one pixel of 200 nm. In this experiment, an average number of 340 counts per molecule was obtained, concomitant with a localization precision of 22 nm, indicated by the radius of the circles. The length of the trajectory of six observations was representative throughout our study.

We first analyzed data recorded at 20°C at the bottom membrane of the cells (Fig. 7 A; see also Table 2). The mean square displacement increases linearly with the time-lag, as expected for free Brownian motion. A fit according to Eq. 1 yields a diffusion constant D = 0.30 ± 0.02 μm²/s. No saturation or offset of the curve can be observed within the error bars ( ). Similarly, no saturation or offset has been detected when measurements were performed at the top membrane, or at 37°C. However, as expected, the latter measurement revealed a higher diffusion constant D = 0.46 ± 0.05 μm²/s (Fig. 7 A). We conclude that on a length scale of the membrane protein diffuses freely; if confinement regions exist, they are either too large or too small to be detected directly in this plot.

TABLE 2.

Summary of all experimental results obtained in this study

	T	till	CO	Dmacro	N	Color
Fab-fragment	37°C	0.30 ms			150	Green
Fab-fragment	20°C	0.65 ms			118	Black
Full Ab	37°C	0.65 ms			132	Red
Full Ab	20°C	0.65 ms			142	Cyan
Full Ab	20°C	0.05 ms			191	Blue

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For each experimental realization, N trajectories were analyzed. CO and D_macro were obtained by fitting MSD versus t_lag according to Eq. 1. All data were corrected for localization precision according to , . The color code relates to Fig. 10.

To test for larger confinements, we repeated the experiments at longer time-lags of t_lag = 15 ms, 50 ms, and 101 ms (Fig. 7 B). MSD was found to increase slightly sublinear with time-lag, an effect that had been observed frequently for membrane proteins (27). Anomalous subdiffusion is generally ascribed to molecular crowding and transient binding (28–30), and can be described by , with d the anomalous diffusion exponent. The measured value is close to unity, showing that the mobility over long timescales was decreasing only slightly. For the applied conditions there is no indication for confinement of CD59 to regions with a size .

We next tested whether confinements could be introduced by transient cross-linking of two CD59 molecules. We therefore repeated the above experiments using the full antibody to CD59. This antibody had the additional advantage that it could be labeled with on average more than one Alexa647 molecule without losing its binding affinity. We made use of this advantage by reducing the illumination time to 50 μs; during such short illuminations, the molecules are practically immobile, which allows for determining the correct position without influences due to positional averaging according to Eq. 3. Fig. 8 shows the results of the analysis: although the diffusion constant is decreased to D = 0.11 ± 0.03 μm²/s at 20°C (D = 0.17 ± 0.01 μm²/s at 37°C) due to the motion of a larger object in the plasma membrane (31), no significant offset was detected.

It could still be possible that only a fraction of molecules undergoes confined diffusion, which is masked by the majority of nonconfined molecules. We therefore investigated the statistics of the diffusion process by measuring the probability distribution P(δr², t_lag) of individual square displacement steps; this function specifies the probability that a particle starting at the origin will be found within a circle of radius δr at time t_lag. Although for free Brownian motion of a single component a monoexponential function is expected according to , components with a different mobility would contribute as additional fractions with different characteristic diffusion lengths r₀ (20). Fig. 9 shows the according distribution of the data displayed in Fig. 7 A, 37°C: no indication for additional components was found. In addition, all other recorded data could be fitted perfectly with a monoexponential function (supplemental Figs. 4 and 5 and data not shown).

DISCUSSION

In this study, we addressed the question whether confinements restrict the Brownian motion of CD59—a GPI-anchored protein—in the plasma membrane of living T24(ECV) cells. From photobleaching experiments reported in the literature, we know that this molecule is diffusing freely on length scales >1.4 μm (32). However, it has been shown recently that the movement of gold-labeled lipids or lipid-anchored proteins in the exoplasmic leaflet of the cellular plasma membrane is hindered by immobile periodic structures, which confine the diffusing membrane constituents to corrals with a size of ∼120 nm (8,9). It was the aim of this study, to confirm or amend the results of those experiments with a complementary method under less invasive conditions.

We used fluorescent Fab fragments to CD59 for labeling, and recorded the trajectories of single protein molecules at maximum resolution in space and time. To achieve best localization precision, the signal brightness had to be maximized. Increasing the illumination time for this purpose, however, was impracticable since residual movements during illumination would have strongly affected the shape of the recorded trajectories, resulting in an underestimation of confinement sizes (18). We therefore selected a dye with high extinction coefficient ɛ = 220,000 M⁻¹ cm⁻¹ at 647 nm and high saturation intensity , which can be rapidly cycled between ground state and excited state. Using Alexa647 and employing a high excitation intensity I = 44 kW/cm², on average 340 photons have been detected from a single molecule during an illumination time of only 0.6 ms. In addition to signal brightness, the image pixelation and background noise define the precision for localizing single molecules (25,26): for optimum performance, a pixel-size equal to the standard-deviation of the point-spread function of 200 nm was selected (33). Finally, by choosing long wavelength excitation autofluorescence background could be dramatically reduced; for t_ill = 0.6 ms, a background noise b = 6 counts was measured. The precise value of the localization precision was determined from single molecule trajectories recorded on fixed cells for different illumination times; it was in excellent agreement with theoretical predictions (26). Experiments have been performed at settings in a range of 22 nm < σ_xy < 30 nm.

Single molecule observations are terminated by photobleaching, which sets a limit to the length of individual trajectories (34). The application of reducing agents to increase the stability of fluorophores was not feasible here, as such reagents are toxic to cells. We therefore did not attempt to image confined diffusion directly at the single molecule level, but to detect its fundamental consequence: an increase in average jump distances due to the hypothetical rapid random walk within a corral—here termed confinement offset CO—which adds up to the macroscopically observed slow diffusion. A similar consequence has been recently suggested as promising observable for analyzing confined diffusion using fluorescence correlation spectroscopy (35,36). On the millisecond timescale the confinement offset solely depends on the corral size L and the confinement strength ; an analytical approximation has been derived as basis for estimating the confinement strength. Maximum sensitivity of the assay is obtained at high frame rates, where the contribution of CO to the observed jump distances becomes large compared to the macroscopic slow diffusion. We therefore employed the kinetics mode of the CCD camera to record images at acquisition rates of up to 1000 frames per second.

Let us assume corrals of 120 nm size as reported for T24(ECV) cells by single particle tracking (9). Within the bounded region, CD59 shall move rapidly with a diffusion constant of D_micro = 8 μm²/s (9); in this case, the particle would diffuse freely for a time , too fast to be resolved with single molecule methods. For the experiment shown in Fig. 7 (37°C), however, according to Eq. 3 we would expect a confinement offset , which was not detected here (). Even more so, variation of the temperature or usage of full antibodies to induce dimerization had no significant effects on the confinement offset (see Table 2).

The resolution of single molecule microscopy reaches its limits in the presented study. Residual errors in determination of CO might mask the effect of periodic permeable barriers. To provide an unambiguous conclusion, we performed an error analysis based on the analytical description of hop diffusion (Fig. 10). Apparently, the value of CO depends both on L and , which can be regarded as the only unknowns in the model (Eq. 3). In particular, for small values of CO would reduce to zero, which renders a precise measurement of the confinement size in this parameter regime difficult and prone to errors. In contrast, high values of are concomitant with high values of the hypothetical microscopic diffusion constant D_micro; in this regime, results are sensitive to positional averaging effects according to Eqs. 2 and 3. It is therefore not possible to determine L and independently by this method. However, we can provide an upper boundary for the parameter duplet by accounting for all potential sources of errors. As described in the error analysis section, we first estimated the error of each measured parameter (, , ), calculated via Gaussian error propagation analysis as a function of L and , and compared this value with the expected confinement offset ; the equality therefore defines a region bounded by the maximum values of the duplet , which are consistent with the data within a one-sigma confidence limit (84% accuracy). For each experimental realization (variation of temperature, illumination time, full antibody versus Fab fragment) we determined the maximum values of the duplet , which would yield consistency between the measured data and the hypothesis of square confinement regions in the plasma membrane (Fig. 10). In other words, the lower left region of the plot bounded by the set of curves represents the parameter range, in which confinement regions cannot be excluded within the one-sigma confidence limit. It has to be noted that for each data a second duplet exists that provides a solution to the equation . However, this duplet specifies a lower boundary to , yielding unrealistically high values of D_micro and has thus not been taken into further account.

For large values of L the curves approach , which is equivalent to unconfined diffusion; in other words, the existence of large but highly permeable corrals would be in agreement with the data. This is not unexpected, as with deviations of MSD from a linear increase vanish. In contrast, high values of are concomitant with the extreme case of strong confinement, which directly provide an estimate for the maximum value of L consistent with the data. The spread within the curve set reflects the different limitations of the applied conditions: e.g., although achieving highest localization precision using long illumination times, positional averaging due to residual motion during the illumination renders this data less strict. This effect can be observed when comparing the blue curve representing experiments at t_ill = 50 μs with the remaining curves recorded at 10 times longer illuminations. At t_ill = 50 μs, diffusion during illumination hardly affects the observed confinement offset, therefore the curve is rather flat for small values of L.

The presented analysis has been based on a homogenous population of molecules moving in a perfectly periodic meshwork of barriers. Let us estimate the consequences of deviations from this model. One might speculate that not all molecules were confined, with a large number of freely diffusing molecules masking the confinement effect. This scenario, however, is not supported by our data, as statistical analysis did not indicate any additional component with a distinct diffusion behavior (Fig. 9). Moreover, the expected high value of D_micro ∼ 8 μm²/s would be inconsistent with the measured D_macro = 0.46 μm²/s. Let us next assume corrals to be not perfectly periodic, with considerable variations in size. A distribution of L values would yield an average MSD according to , with p(L²)dL² the probability of finding a domain with area in an interval [L², L² + dL²]. Since , analysis of the data would yield a proper estimate of the average domain area, but a too large estimate of the domain size, resulting even in an overestimation of L. Monte Carlo simulations performed on domains with broad size-distributions corroborate the interpretation (data not shown).

We can interpret the obtained results in two different ways: first, strong confinements with exist in T24(ECV) cells, however, with a size much smaller than reported in the literature (9). In this scenario, our data provide strong evidence that such corrals have to be smaller than ∼60 nm, at 20°C even smaller than 40 nm. We want to point out, however, that there is no report in the literature for the existence of confinements in T24(ECV) cells with such a size, rendering this scenario rather speculative. Second, periodic restrictions to the free Brownian motion of CD59 in T24(ECV) cells exist with the reported value of L = 120 nm (9), but have a low confinement strength . In this case, the barriers are hardly sensed by the moving molecule, leading to no significant decrease of D_macro compared to D_micro.

In conclusion, our data clearly indicate that the motion of the GPI-anchored protein CD59 is not restricted by periodic cytoskeletal barriers. At this point, it is difficult to reason why the presented approach did not yield the same results as previous studies performed with larger particles as labels. It seems as if gold labeling leads to an exaggerated confinement strength, maybe due to its ability to transiently bind specifically or unspecifically to other membrane proteins. Single particle tracking of gold-labeled membrane constituents may still provide a proper method to probe diffusion barriers; however, our results show that the single molecule hop frequency and concomitantly the confinement strength cannot be directly inferred from single particle tracking studies.

SUPPLEMENTARY MATERIAL

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