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Mol Cell Biol. Jan 2006; 26(1): 313–323.
PMCID: PMC1317633

Identifying Optimal Lipid Raft Characteristics Required To Promote Nanoscale Protein-Protein Interactions on the Plasma Membrane

Abstract

The dynamic lateral segregation of signaling proteins into microdomains is proposed to facilitate signal transduction, but the constraints on microdomain size, mobility, and diffusion that might realize this function are undefined. Here we interrogate a stochastic spatial model of the plasma membrane to determine how microdomains affect protein dynamics. Taking lipid rafts as representative microdomains, we show that reduced protein mobility in rafts segregates dynamically partitioning proteins, but the equilibrium concentration is largely independent of raft size and mobility. Rafts weakly impede small-scale protein diffusion but more strongly impede long-range protein mobility. The long-range mobility of raft-partitioning and raft-excluded proteins, however, is reduced to a similar extent. Dynamic partitioning into rafts increases specific interprotein collision rates, but to maximize this critical, biologically relevant function, rafts must be small (diameter, 6 to 14 nm) and mobile. Intermolecular collisions can also be favored by the selective capture and exclusion of proteins by rafts, although this mechanism is generally less efficient than simple dynamic partitioning. Generalizing these results, we conclude that microdomains can readily operate as protein concentrators or isolators but there appear to be significant constraints on size and mobility if microdomains are also required to function as reaction chambers that facilitate nanoscale protein-protein interactions. These results may have significant implications for the many signaling cascades that are scaffolded or assembled in plasma membrane microdomains.

The classical view of the plasma membrane lipid bilayer, as a two-dimensional fluid acting as a neutral solvent for membrane proteins in which all particles diffuse freely (34), has been substantially modified in recent years. The plasma membrane is a complex structure that is compartmentalized on multiple length and time scales. This compartmentalization is driven by a variety of lipid-lipid, lipid-protein, and actin cytoskeleton interactions (1, 5, 11, 13, 17). These mechanisms are well illustrated by the plasma membrane interactions of the Ras GTPases that have been extensively studied by single-particle tracking (SPT), electron microscopy (EM), and fluorescence recovery after photobleaching (FRAP). H-, N-, and K-ras are nonrandomly distributed over the plasma membrane in microdomains or nanoclusters. Activated GTP-loaded H-ras and K-ras dynamically partition into nonoverlapping clusters that are unaffected by cholesterol depletion but exhibit differential dependence on the actin cytoskeleton (16, 18, 24, 25). Activated GTP-loaded N-ras, in contrast to H-ras, partitions into cholesterol-dependent clusters (8, 16, 26). Clustering of H- and K-ras proteins requires the common scaffold protein Sur-8 and the selective scaffolds galectin-1 and galectin-3, respectively (6, 8, 16, 21). There is good evidence that these various Ras microdomains are platforms for signal transduction (8, 16, 26).

A further example of membrane microdomains is the lateral segregation of glycosphingolipids and cholesterol into liquid-ordered domains. Phase separation of cholesterol-enriched, liquid-ordered domains or lipid rafts has been clearly demonstrated in model membranes and also in biological membranes, although the length and time scales on which this phase separation occurs are a matter of debate (5, 11, 15, 32). Multiple estimates of the diameter of lipid rafts have been provided using diverse techniques, although photonic force microscopy, fluorescence resonance energy transfer, and EM provide a convergence of estimates of 6 to 50 nm, with the most recent studies favoring the lower end of this range (5, 23, 24, 31). Similar sizes, in the range of 12 to 32 nm, have been reported for the microdomains occupied by activated H-ras and K-ras (22, 24).

An important role that has been ascribed to all plasma membrane microdomains is that of selectively concentrating proteins to facilitate the assembly of signaling complexes (33). Many studies have been qualitatively interpreted in terms of this type of microdomain model. However, no quantitative analysis has been attempted to explore the basic mechanics of how microdomains might drive protein-protein interactions, as demanded in their role of supporting the assembly of signaling platforms. For example, if microdomains do aggregate proteins, are there any constraints on size and dynamics that need to be imposed for them to achieve this function? If so, are these constraints realistic and how do the predictions compare with recent estimates of microdomain size and dynamics? These are difficult but important questions that are especially relevant in the context of the ongoing discussions of plasma membrane structure and function.

Since microdomains are larger than single proteins, the diffusion rate of proteins sequestered in microdomains is lower than those in surrounding membranes (23). However, if proteins enter and leave microdomains, i.e., exhibit dynamic partitioning, the expected local and global effects on protein diffusion are more difficult to predict. The purpose of this study is to explore these basic issues by constructing and interrogating stochastic Monte Carlo models of the dynamics of protein molecules on a membrane in the presence of microdomains. For the purpose of this paper, we will consider lipid rafts as a specific example of a plasma membrane microdomain, given the extensive literature that evokes these structures to explain biological observations and the desire to provide an appropriate computational approach for evaluating these claims.

MATERIALS AND METHODS

Model description and Monte Carlo simulations.

The model relies on the representation of the cell membrane as a two dimensional lattice. Each element of this lattice is a “voxel” that can be either occupied or unoccupied by a modeled protein at each time step; in the former case, a record is made of which modeled protein occupies the voxel. At any time, only one modeled protein may occupy a given voxel, in order to ensure volume exclusion between modeled proteins. For brevity, we will subsequently refer to “modeled proteins” simply as “proteins.” Throughout this work, the size of a voxel has been assumed to be 2 nm by 2 nm, this being an estimate of the average size of a membrane-anchored protein (and not the size of the anchor). The lattice size for simulations emulating FRAP experiments (see below) is 5 μm by 7.36 μm (9.2 × 106 voxels), while in all other simulations it is 500 nm by 736 nm (9.2 × 104 voxels). The two main considerations in choosing the voxel size are (i) the size of a membrane-anchored protein (so that volume exclusion can be accurate) and (ii) the dimensions of the membrane (must be large enough to get accurate statistics but small enough to make the simulations tractable).

A lipid raft is modeled as a two-dimensional patch whose boundaries are described by a simple closed curve in the plane. In this work, rafts are represented as disks and so a modeled raft occupies an area in the plane Aξ = πrξ2 and is surrounded by nonraft regions as well as other modeled rafts. A given voxel is thus assigned to either a raft or a nonraft region. In this study, modeled rafts can have various radii, different densities on the membrane, and different diffusive properties but, in any given simulation, the parameters remain fixed. Boundary conditions are periodic. For brevity, we will subsequently refer to “modeled rafts” simply as “rafts.”

The lattice is seeded with proteins of different species [for each species i, let the number of proteins present in the system initially be ni(0)]. Each protein has two properties: a position specified in terms of its x and y coordinates in the lattice and a species. At each step, a protein M is chosen at random from the general population. Let the coordinates of this protein be (x,y). One of the voxels with coordinates (x + Di,y), (xDi,y), (x,y + Di), or (x,yDi) is also chosen at random, where Di (a positive real number) is the step size of species i that depends on its location (inside or outside a raft). The new voxel is the location to which Brownian motion moves the protein during the current time step alone. Note that the case where D equals 1 corresponds to choosing an adjacent voxel. If the new voxel is occupied by a protein or a fixed obstacle, then protein M is placed back in its original voxel (x,y) and a collision is recorded. This is an implementation of volume exclusion, so that only one protein can occupy a voxel at any time, as described above (3). After each such step, the simulation time is incremented by 1/n, where n is the total number of proteins on the membrane. Thus, a simulation interval of 1 corresponds (statistically) to the time needed for all proteins to move once.

The “baseline” step size of a protein in a nonraft region is set to D equals 1, and because rafts move more slowly than proteins, the step size of a raft is less than unity. These values are estimated from the work of Saffman and Delbruck (27) and are shown in Table Table1.1. Since the mobility of a protein in a raft is reduced compared with that of the surrounding membrane region (4, 23, 30), an important parameter that has been used throughout this study to describe the interaction of a protein with a raft is the ratio of the step size of a species inside and outside a model raft ρi.

equation M1
(1)

Larger values of D correspond to higher diffusion rates, that is, a better-mixed system. If D is 0, then the raft or molecular species in question is immobile. If D is nonintegral, then the interpretation of D is probabilistic and the size of the diffusive step is nondeterministic. For example, if Di is 0.5, then a protein of species i has, at each step, a probability of 0.5 of moving to one of its neighboring voxels (if unoccupied) and an equal probability of not moving at all. This is used to implement statistically subvoxel step sizes (the unitary step size must always be the size of one voxel, 2 nm). Throughout this study, we have used a step size of 1 voxel outside rafts and a step size of ρ inside rafts. Because the membrane size is large relative to a voxel size, the algorithm is a good random walk approximation to Brownian motion.

TABLE 1.
Step sizes of rafts relative to proteinsa

The differing numbers of rafts and proteins in the simulation complicate the modeling somewhat. If a raft is chosen at random during each Monte Carlo step and moved to a new position, the diffusion coefficients of rafts and proteins cannot be equated numerically. As an example, consider a situation where there are 10 rafts and 100 proteins. A protein will move, on the average, once every 100 simulation steps; using a naïve implementation, each raft will have moved 10 times at the end of that period since it has a 1/10 probability of being chosen during each of the 100 steps. This is not a physically accurate realization: each raft should only have moved once, to keep diffusion rates independent of time. To achieve this, at each Monte Carlo step, the motion of rafts is governed by two random numbers. The first of these, R1, is sampled from the uniform distribution on [0, 1] and is used to decide whether or not any raft will move during this step. Specifically, if R1 is smaller than the raft-to-protein ratio Rr/m defined by nrafts/nproteins, where nrafts and nproteins are the total numbers of rafts and proteins present, respectively, then one raft will move during this step; otherwise, all rafts retain their positions. If it is decided to move a raft, a second integer, R2, is chosen at random between 1 … nrafts and used to decide which raft will be moved. That raft will be moved in one of the cardinal directions, as for proteins and according to its step size. If, however, the new area to be occupied by the raft overlaps with that occupied by another raft, then it is returned to its old location. Thus, a volume exclusion condition is imposed between rafts. The parameters tracked during simulations are (i) the proportion of proteins in rafts, (ii) the equilibrium concentration of proteins in rafts after allowing the system to reach a steady state, and (iii) the collision rate of proteins. A collision between two proteins is recorded whenever one of the two attempts to move to a voxel that is occupied by the other (and is rejected). The time derivative of the total collision count, estimated using a difference formula, is the approximate number of collisions per unit time at each point in the simulation.

To simulate an actin picket fence, the membrane was partitioned by an array of fixed obstacles. In the framework of the model, obstacles are represented as inert proteins with a step size of 0. Proteins attempting to move into a voxel occupied by an obstacle are rejected. Similarly, no portion of a raft is permitted to move over a voxel containing a fixed protein (obstacle); this is biophysically realistic since rafts are postulated to be tightly packed lipid-cholesterol complexes that are considerably less fluid than the surrounding membrane.

To simulate raft affinity, “rejection probabilities” were introduced associated with proteins entering and exiting a raft. When a protein moves from a nonraft voxel to a raft voxel, it may be returned to its original location (rejected) with probability prr. Conversely, when exiting a raft, it may be rejected with probability prn. If these probabilities are 0, the proteins do not differentiate between rafts and nonraft regions, except for the difference in diffusion rates they experience (i.e., raft partitioning is always influenced by ρ irrespective of the value of prn). At the other extreme, a probability of prn or prr equal to 1 indicates that once a protein has entered a raft or nonraft region, respectively, it will be permanently captured in that region.

Anomalous diffusion.

A consequence of the presence of rafts in our model is the reduced mobility of proteins overall, especially if the rate of diffusion of these inside raft domains is significantly lower than those in the free membrane. In order to measure this, the average diffusion coefficient of proteins within a simulation is calculated in the following manner. The mean squared deviation <X(t)2> of a protein from its starting voxel is recorded. This obeys the anomalous diffusive relationship (14)

equation M2
(2)

where D is the diffusion coefficient, Γ(k) is the gamma function defined as

equation M3
(3)

when k is >0, and α is the anomalous exponent, a measure of the nonclassical behavior of the Brownian motion executed by a particle (a protein in this case). Thus, the coefficient of diffusion can be estimated from the vertical intercept y of a straight line fitting the log(<X(t)2>)-log(t) data

equation M4
(4)

and α is estimated from the slope of the same line. Note that in the case of pure diffusion, α = 1 and <X(t)2> = 2Dt (a linear relationship).

FRAP simulations.

To simulate FRAP experiments, all proteins are given a “tag” property that has value of 1 if they are fluorescent and 0 otherwise. At the beginning of the simulation, all proteins have tag values of 1. When the system has reached equilibrium with respect to spatial distributions of proteins, all proteins in a particular area of the membrane have their tags set to 0 (while all proteins outside this area have unchanged tags). Subsequently, the total sum of the tags over the “bleached” area is recorded periodically and this procedure is repeated until this sum, representing the total fluorescence due to the bleached area, returns to its initial value. From this data, t0.5, the time required for the fluorescence signal to return to half of its initial value is extracted (2, 9). The relationship between the diffusion coefficient and t0.5 is

equation M5
(5)

where Dmacro is the large-scale diffusion rate, ω is the bleach radius, and γ is a correction factor (0.88 for a circular bleached area). The “macro” subscript refers to the fact that because of the relatively long time scales involved in FRAP experiments (and our simulations), the diffusion coefficient estimated using this method reflects the large-scale mobility of proteins. In the case of pure diffusion, one would expect the diffusion coefficient to be independent of the time and space scales (it is a constant in the diffusion equation). Recent studies, however, have suggested that diffusion is strongly impeded over large space scales so that the short-range diffusion coefficient Dmicro is not, in fact, equal to the large-scale coefficient Dmacro (28, 29). We estimated both Dmicro and Dmacro using direct short time scale and FRAP simulations, respectively, in an effort to investigate whether the presence of rafts accounts for the nonconstancy of the diffusion coefficient.

RESULTS

Constructing a Monte Carlo model of the plasma membrane.

Here we establish a quantitative framework for examining how lipid rafts influence plasma membrane protein dynamics. We consider rafts in the context of their classical representation as domains enriched in cholesterol and sphingolipids that diffuse as stable entities within the fluid bilayer (23, 33). Although we refer specifically to lipid rafts, it should be stressed that the model is generally applicable to any type of plasma membrane microdomain that diffuses as a stable entity within the fluid bilayer, irrespective of whether the integrity of the microdomain is cholesterol dependent. Membrane proteins exist in three categories: class 1, raft associated, being present in mainly raft domains; class 2, nonraft associated, being present in mainly the liquid-disordered phase; or class 3, dynamic partitioning proteins that move in and out of rafts; although classes 1 and 2 are simply extreme representations of class 3. If rafts are mobile, a protein in a raft will be acted on by two independent sources of motion: its own diffusive motion and the diffusion of the raft that contains it. If a protein is excluded from rafts, its mobility and diffusion will also be influenced by the sea of rafts in which it operates.

A Monte Carlo approach is particularly suited to the problems considered here because the local motions of proteins and rafts can be implemented using simple transition rules, while individual events, such as collisions between proteins, can easily be recorded in time. A detailed description of the model is provided in Materials and Methods. In brief, a model lattice membrane was constructed onto which proteins and rafts were randomly seeded. The proteins and rafts were then moved randomly by Brownian movement. The area of the membrane that was designated lipid raft and the radius of the rafts were varied over wide ranges but were fixed during any given simulation. In the model, rafts move more slowly than individual proteins because of their greater cross-sectional diameter. The relative velocity of rafts of a given radius compared to that of an individual protein was estimated using the Saffman-Delbruck equation (27) and is shown in Table Table1.1. The mobility of a protein in a raft is reduced compared with that of the surrounding membrane region because of the dense packing of lipids in the liquid-ordered raft environment (4, 23, 30). This critically important behavior is reflected in the parameter ρi, which is the ratio of the distances moved by a protein in unit time, inside and outside a raft (see equation 1 in Materials and Methods). The influence of this parameter on the behavior of protein diffusion and protein-protein interactions forms the first part of this study.

To interrogate the model, a number of parameters were tracked for each simulation as follows: (i) the proportion of proteins in rafts, (ii) the equilibrium concentration of proteins in rafts after allowing the system to reach a steady state, and (iii) the collision rate of proteins. The collision rate was chosen as a simple and qualitative first approach to probing the effects of lipid rafts on cell membrane chemistry, although chemical reactions were not considered explicitly in this work.

To fully characterize the properties of these model rafts, we first examined in detail the situation where rafts have no inherent affinity for a protein but operate simply under the parameter ρ. These simulations therefore reflect the behavior of proteins that interact with the plasma membrane primarily via N- or C-terminal lipid anchors. If the diffusion rate of a protein is lower in rafts, then, intuitively, the residence time of a protein in a raft should, on average, be greater than that in a nonraft region of equal dimensions. In other words, proteins are more likely to be found inside rafts than outside them if ρ is <1. This hypothesis was tested with simulations in which the initial distribution of proteins was uniform and random: the dimensions of the membrane were 500 nm by 736 nm onto which 2,500 proteins were placed. Representative results are shown in Fig. Fig.1.1. The proportion of proteins in rafts begins at 25% since, in this simulation, rafts represent 25% of the membrane, and proteins are distributed randomly. As proteins enter rafts through diffusion, they are less likely to escape back to nonraft regions due to the reduced rate of diffusion rate in the raft domains. This leads to an aggregation of proteins in rafts, which becomes more pronounced as the value of ρ decreases. Conversely if ρ is 1, so that proteins are diffusively “blind” to the presence of rafts, the effect is negligible. If rafts are immobile and ρ is 1, there is no aggregation. If rafts are mobile and ρ is 1, a minor amount of aggregation is observed due to the “sweeping up” of proteins by mobile rafts; for example there is an 3% increase in Fig. Fig.1.1. A raft “swallows” proteins that are in its path, and these are subsequently moved along with the raft during their period of residence. Figure Figure11 also shows that, after a transient period during which the proteins are aggregating in rafts, an equilibrium concentration is reached at which the increased residence time in rafts is balanced by the nonzero probability of escape.

FIG. 1.
Example output from a Monte Carlo simulation. The graph shows the fraction of proteins on the membrane that are present in rafts as simulation time increases. The simulations were run to equilibrium. ρ is the ratio of diffusion rates inside and ...

The equilibrium concentration is essentially independent of raft size and mobility.

How does the equilibrium concentration vary with raft dimensions and with ρ? To answer this question, we performed experiments using all combinations of the following parameters: (a) total raft area equaling 10%, 25%, and 50% of the membrane, (b) raft diameters equaling 6 nm, 14 nm, 26 nm, and 50 nm, (c) rafts immobile and mobile, and (d) ρ equaling 0.25, 0.5, 0.75, and 1. The membrane area and number of proteins are the same as in Fig. Fig.1,1, and the simulations were run to equilibrium. A complete set of results is shown in Fig. Fig.2.2. The most striking feature of these results is that the equilibrium concentration is essentially independent of raft diameter and is not greatly affected, overall, by raft mobility. The equilibrium concentration of proteins in rafts falls, albeit quite weakly, with increasing ρ; as ρ→1, this concentration tends to the proportion of the membrane that the rafts occupy, as would be expected since, when ρ equals 1, rafts are either blind (in the immobile case) or almost so (in the mobile case) to the presence of rafts. Conversely, at low values of ρ, the probability of escaping from a raft is low and the aggregation effect is substantial, leading to a high equilibrium concentration in rafts. Thus, as ρ→0, the rafts act increasingly as “protein sinks.” The dependence of equilibrium protein concentration on ρ appears moderately stronger as total raft area decreases, but this is so simply because the initial concentration in rafts then also decreases. We conclude, therefore, that the equilibrium concentration of a protein with no inherent affinity for rafts is predominantly independent of raft dimensions and raft mobility but moderately and approximately linearly dependent on the difference in diffusion rates of that protein between raft and nonraft regions.

FIG. 2.
Equilibrium concentrations of proteins in lipid rafts. As in Fig. Fig.1,1, proteins have no intrinsic affinity for rafts; association is driven by the parameter ρ. Simulations were run for four values of ρ, with rafts occupying ...

Rafts have modest effects on (small-scale) diffusion rates of proteins.

We next investigated to what extent the average diffusion rate of proteins is influenced by the presence of rafts. The diffusion coefficient was estimated using equation 2 from the mean squared displacement data recorded during the simulations. Figure Figure33 shows that the diffusion coefficient grows with ρ, and the dependence is approximately linear for immobile rafts and is not strong but increases somewhat with raft area. If rafts are mobile, the dependence is more tenuous and reveals a complex interplay between raft size (and hence diffusion rate), ρ, and total raft area. In the immobile case, the diffusion coefficients converge as ρ→1, while in the mobile case they do not, due to differences between the mobility of differently sized rafts (Table (Table1).1). The diffusion coefficients in the presence of mobile rafts are always greater than when rafts are immobile, but the separation is small and only fully resolved when the raft area is ≥25%.

FIG. 3.
The coefficient of diffusion of proteins in the presence of lipid rafts. As in Fig. Fig.1,1, proteins have no intrinsic affinity for rafts; association is driven by the parameter ρ. Simulations were run for four values of ρ, with ...

Figure Figure33 shows that the plots of D versus ρ for immobile rafts of different diameters are superimposed on each other. Thus, if rafts are immobile, raft diameter has no effect on the diffusion coefficient. In contrast, the plots of D versus ρ for mobile rafts do show some weak separation at high raft areas with 6-nm rafts slowing diffusion when ρ equals 0.25 to a greater extent than larger rafts. We conclude that rafts have only a moderate effect on the diffusion coefficients of raft-partitioning proteins. In the most extreme realization of the raft model, where 50-nm diameter immobile rafts occupying 50% of the cell surface and ρ equals 0.25 so that 90% of the proteins are partitioned in the rafts at any given time point (Fig. (Fig.2),2), diffusion would be slowed only ~2.5-fold compared to the same protein diffusing in a plasma membrane containing no rafts (Fig. (Fig.3).3). For a single protein diffusing on a free membrane, the maximum value of D is exactly 0.5. Note, however that in some simulations with mobile rafts, D is higher than this value. This phenomenon is perhaps due to the combination of the two independent movements of a protein: (i) Brownian motion and (ii) the movement of the raft in which the protein is embedded. These two effects may be roughly additive, resulting in the observed diffusion rate of a protein being higher than 0.5.

Collision rates between proteins are maximal when rafts are small (≤14 nm) and mobile.

We tracked the total number of collisions between proteins occurring inside and outside of rafts in all of the simulations summarized in Fig. Fig.22 and and33 and calculated a collision rate as described in Materials and Methods.These results show that if rafts are immobile, the collision rate is insensitive to both raft size and the parameter ρ (Fig. (Fig.4).4). In a sense, this is not unexpected. Although the concentration of proteins in rafts is higher than in the surrounding membrane, leading potentially to a higher rate of collision, this is offset by the lower diffusion coefficient in rafts causing reduced protein mobility and hence reducing the rate of collisions. However, this is not the case if rafts are mobile. Simulations show that the collision rate depends on ρ, reaching a maximum at around ρ equals 0.5 and falling away at smaller and larger values. The collision rate also varies strongly with raft size but in a complex interplay with raft area. Thus, 6-nm mobile rafts allow the highest collision rates when the raft area is 10% but allow the lowest when raft area is 50%. In contrast, a raft diameter of 14 nm maximizes collision rates at areas >25% and provides close to maximal collision rates at 10% area. We speculate that the increase in collision rate observed when rafts are mobile is due to mainly collisions between rafts, which bring their constituent proteins (especially those close to the rims of the rafts) in close proximity. We conclude that if the function of rafts is to facilitate protein-protein interactions, then raft mobility is critical to realizing this function. Moreover, the model suggests that an optimal diameter for rafts to maximize collisions between raft-partitioning proteins is ~14 nm. The maximum collision rate is reached when ρ equals approximately 0.5, which represents a good compromise between increased concentration of proteins in rafts and reduced protein mobility.

FIG. 4.
Collision rates in the presence of lipid rafts. Total collision rates between proteins were recorded during each of the simulations summarized in Fig. Fig.22 and and3.3. As in Fig. Fig.1,1, proteins have no intrinsic affinity for ...

Picket-fence experiments.

Given that the data presented earlier suggests that the effect of rafts on protein mobility on the small scale (Dmicro) is not large, we also tested the effect of placing an “actin fence” on the membrane. In accordance with reference 7, the “pitch” of the fence array was set to 100 nm and the perimeter density (i.e., the density of fixed obstacles) was set to 20%. The raft dimensions were set to 14 nm, ρ to 0.5, with a raft coverage area of 50%, and simulations were run for 500 time steps. The effect of the fence on mobile and immobile rafts was tested because, as described in Materials and Methods, collisions with the fence impede the movement of mobile rafts. The results are shown in Table Table22 and compared with earlier data in which there was no fence post arrangement. We find that the presence of the fence has an insignificant effect on the collision rate of proteins if rafts are fixed but moderately decreases the interprotein collision rate if rafts are mobile. The diffusion rate increases very slightly in the presence of a fence system, and the equilibrium proportions of proteins in rafts fall by about 10%. We conclude that the presence of a picket fence system characterized by the parameters above has a relatively minor effect on the dynamics of the system.

TABLE 2.
Effects of an actin fencea

In silico FRAP experiments.

FRAP is frequently used to estimate the lateral diffusion of plasma membrane proteins over large distance and time scales. A commonly held and perhaps intuitive expectation is that the lateral diffusion rate of raft-associated proteins is significantly lower than nonraft proteins because rafts diffuse slower than single proteins. The analysis presented in Fig. Fig.33 suggests, however, that the diffusion coefficients might, in fact, be expected to vary by only <2.5-fold even in a most extreme realization of the raft model. To compare the actual diffusion rates measured in Fig. Fig.33 with the results of a FRAP experiment, we simulated FRAP by first allowing the distribution of proteins on the membrane to reach steady state over 500 simulation time units and then “photobleaching” a circle with a diameter of 500 nm. The total area of the membrane was increased to 5 μm by 7.36 μm, so that the number of bleached proteins was <0.6% of the total number of proteins. This ensures that the fluorescence signal can recover fully after “photobleaching.” We ran simulations for four sets of parameters: (i) no rafts, (ii) mobile rafts of 14 nm diameter making up 50% of the membrane with (ρ = 0.5), (iii) mobile rafts of 14 nm diameter making up 25% of the membrane with (ρ = 0.5), and (iv) mobile rafts of 14 nm diameter making up 25% of the membrane but with proteins excluded from rafts (i.e., if a protein attempts to enter a raft region, it is rejected and does not move from its current voxel). Figure Figure55 shows that the fluorescence signal recovers much more slowly if rafts are present. For example, the half-recovery time is around 1,950 time steps if no rafts are present, while if rafts represent 50% of the total membrane area, half-recovery time is around 25,500 time steps. Calculation of the large-scale diffusion coefficients (Dmacro) from these data (Table (Table3)3) shows that these differences correspond to an approximately 3.2-fold increase in Dmacro if rafts are absent. Thus, although the local diffusion rate of proteins is not substantially altered under these two realizations of the model (a 1.2-fold increase if rafts are absent), their long-range mobility is significantly impeded by the presence of rafts. Moreover, Table Table33 also shows that decreasing raft area from 50% to 25% results in only a 1.2-fold increase in Dmacro; thus, a substantial change in the area of the membrane designated raft translates into a relatively modest change in Dmacro. We next examined if a simple FRAP experiment could discriminate between a raft-associated and raft-excluded protein under a realization of the raft model where 25% of the membrane is raft. Figure Figure55 and Table Table33 show the interesting result that exclusion from rafts results in only a minimal, 1.07-fold faster Dmacro than partitioning into rafts. A comparison of the calculated Dmacro from the FRAP data with the measured Dmicro from the simulations shows that, whereas the two coefficients trend in the same direction, their relationship to each other, both in the presence and absence of rafts, is highly variable (Table (Table3).3). To further explore the reasons for the different ratios predicted by the measured diffusion coefficients and the estimated values from the FRAP experiments, we observed the recovery directly in QuickTime movies. For two movies showing recovery of fluorescence for (a) no rafts and (b) 50% rafts with ρ equaling 0.5, see the supplemental material. These indicate clearly the difficulty of fluorescent proteins in entering the bleached area and that of bleached proteins in exiting it, respectively, when rafts are present.

FIG. 5.
In silico FRAP experiments. Four FRAP experiments are shown on a model plasma membrane in which rafts are not present; rafts cover 50% of the total area, are mobile, have 14-nm diameters, and slow down the diffusion of proteins by a factor of 0.5 (ρ ...
TABLE 3.
Long range diffusion is impeded by raftsa

Can raft affinity facilitate protein-protein interactions?

To date, we have considered ρ as the sole parameter for driving raft association. We next investigated how an alternative mechanism of confining proteins to raft (or nonraft) regions affects the dynamics of the population. To this end, we distributed 1,000 proteins on the plasma membrane and compared the number of collisions when a protein is unable to escape from rafts once it has entered (prn = 1) with the number of collisions when there is no imposed confinement (prn = 0). The value of the parameter prn gives the probability that a protein attempting to leave a raft will be rejected; it is a simple realization of raft affinity. If raft affinity is high, then prn is 1 and, if raft affinity is low, then prn is 0 (for details, see Materials and Methods). Since prn is a separate parameter from ρ, we evaluated the effect of raft affinity for two values of ρ.

First, to examine the effect of prn in isolation, ρ was set to 1 so that the diffusion rate in rafts is the same as outside. The simulations were run with 14-nm mobile rafts occupying 10 to 50% of the plasma membrane. Figure Figure66 shows that increasing raft affinity from prn equals 0 to prn equals 1 substantially increases collision rate. The amplification is greatest (approximately fivefold) when raft area is smallest (10%). Second, we repeated the same set of simulations but with a diffusion rate reduction in rafts also operating on the proteins (i.e., ρ was set to 0.5). Interestingly, Fig. Fig.66 shows that, under these conditions, there is no significant increase in collision rate by increasing raft affinity, except again when raft area is low. Taking these results together, we conclude that increasing raft affinity (prn) may be an efficient mechanism for concentrating proteins but, in terms of promoting protein-protein interactions, raft affinity is not necessarily additive with lateral segregation promoted by diffusion retardation (ρ), unless the area occupied by rafts is small (≤10%).

FIG. 6.
Raft affinity as a parameter to increase intermolecular collisions. The collision rate between 1,000 randomly distributed proteins (on a 500-nm by 368-nm membrane) was measured with the reflection parameter for raft affinity (prn) set to 0 or 1 (the minimum ...

DISCUSSION

In this study, we have devised and interrogated a stochastic quantitative model of microdomains, such as those exemplified by lipid rafts. We have focused on two fundamental aspects of protein dynamics on the plasma membrane that intuitively should be influenced by the presence of lipid rafts: protein mobility or diffusion and protein collision rates. We have not formally examined the effect of lipid rafts on the velocity of specific biochemical processes, but the analysis of collision rates is a first step in this direction.

The effects of rafts on protein diffusion.

The results presented in Fig. Fig.11 to to33 suggest that the mobility of proteins is not affected to a great degree by the presence of rafts. On the one hand, this is somewhat surprising since the membrane is divided into regions of quite different diffusion characteristics (especially as ρ→0). On the other hand, if we consider the membrane to be a homogenous medium, a proportion pr of which is raft, with ρr being the step size ratio in rafts, then we obtain an estimate for the effective relative diffusion rate (ρeff) over the membrane, namely,

equation M6
(6)

so that ρeff varies linearly with the diffusion ratio. Of course, the approximation of homogeneity is debatable because the rafts are not infinitesimally small. However, as they are small relative to the membrane, it appears that equation 6 is an accurate approximation. In essence, these results indicate that, within the framework of this microdomain model, no significant nonclassical behavior is caused by the presence of rafts on the small spatial scale. Rather, the main effect is to attenuate the global diffusion rate of proteins. Further evidence for this comes from the anomalous exponents computed using the mean squared deviation data (see equations 2 to 4 in Materials and Methods). The case where α equals 1 corresponds to classical diffusion, while α < 1 corresponds to anomalous diffusion, in which the mean squared deviation exhibits a power law relationship in time. In the simulations, the mean value of α over all simulations was found to be 0.973 for both the mobile and immobile raft cases. Minimum values of α were obtained when 50% of the membrane was covered with 6-nm rafts and ρ was 0.25 and when α was 0.86 or 0.85 if the rafts were mobile or immobile, respectively. Values of α in the range 0.8 to 0.9 indicate a small to moderate departure from classical behavior. In any case, ρ = 0.25 is at the low end of the biologically meaningful range of diffusion attenuation factors (estimates of ρ in the range 0.33 to 0.5 have been reported) (4, 23). If ρ is increased to 0.5, the corresponding minimum values of α increase to 0.94 and 0.92, respectively.

Given these results, how are we to explain the FRAP data of Fig. Fig.5,5, showing a very large difference in recovery speeds when rafts are present or absent? The diffusion rate data suggests that the mobility of proteins is not greatly altered by rafts on the short time scale. On long time scales, however, the collisions of rafts with one another reduces their mobility and, since a large proportion of proteins are in rafts, this in turn may prevent proteins from exiting the bleached area. Nonbleached proteins entering the area are likely to collide with proteins already occupying this area, resulting in a low rate of exchange between the bleached and nonbleached regions. These effects can be readily appreciated in the movies in the supplemental material. The results in Table Table33 also show that relating Dmacro to raft area is problematic; decreasing raft area from 50% to 25% results in only a 1.2-fold increase in Dmacro, whereas decreasing raft area from 25% to 0% results in a 2.5-fold increase in Dmacro. In the context of biological experiments that use techniques such as cholesterol-depletion to dissemble lipid rafts, these comparisons become important since they show that quite minor changes in Dmacro could reflect substantial changes in raft area. The data also illustrate the difficulty of using FRAP to discriminate between raft-associated and nonraft-associated proteins. Where 25% of the membrane is raft, exclusion from rafts results in only a 1.07-fold larger Dmacro than partitioning into rafts, again a very small difference. Thus, the seemingly intuitive expectation that because a protein is in a raft it should have a diffusion rate that is significantly lower than a raft-excluded protein is not necessarily realized. These results may explain the similar range of Dmacro values reported for raft- and nonraft-partitioning proteins in a recent comprehensive study (10). Finally the data illustrate that Dmacro deviates significantly from Dmicro. This has been commented on before in various studies on membrane protein diffusion (28, 29), but we show here that this deviation is variable and unpredictable in the presence of lipid rafts.

Interestingly, the presence of an actin fence of low density reduces the equilibrium concentration of proteins in rafts. The reason for this is that, relative to rafts, proteins are considerably more mobile in the presence of even a low-density fence. While rafts are practically totally impeded from crossing into a different membrane compartment, proteins can often slip through the fence. The cytoskeletal fence system does not, however, result in changed overall protein mobility, ruling out, in the framework of this model, notable nonclassical behavior due to such a structure. The fence arrangement at best slightly increases the diffusion rate of proteins (despite the fact that it provides additional obstacles to movement). The picket-fence arrangement, in our model, has only a modest effect on the dynamics of the system because even at high fence-post densities, the primary effect is to reduce the mobility of rafts. In the limit as rafts become completely fixed, the value of D would be reduced by around only 0.1 to 0.15 (see Fig. Fig.33).

The effects of rafts on protein-protein interactions.

An intriguing result from the modeling with significant biological implication is that the collision rate between proteins does not behave in the same way in the cases of fixed and mobile rafts, and, furthermore, the dynamics of the system are different. The lateral diffusion rates of rafts are always less than those of proteins, albeit modestly as shown in Table Table1,1, so that the motions of proteins might be expected to dominate. Nevertheless, the collision rates when mobile rafts are present are almost 1 order of magnitude higher relative to when rafts are fixed or not present (Table (Table2).2). We speculate that the reason for the increased collision rate is that a moving raft, in our model, carries its resident proteins with it. When the raft, through its own diffusion or through external diffusion, captures one or more new proteins, this may result in collisions. As proteins aggregate in rafts, this effect is all the more pronounced. These observations have a clear biological relevance. Caveolae are large (diameter, 65 nm) immobile raft domains that can occupy 4 to 35% of the area plasma membrane according to cell type (19), whereas noncaveolar rafts, at least those occupied by GPI-anchored proteins are smaller (<10 nm) and mobile (31). Segregating proteins in caveolae rather than noncaveolar rafts would, on the basis of our simulations, be predicted to have quite different consequences.

These results suggest a role for rafts as membrane “reaction vessels,” increasing the chance that two specific proteins of low concentration will meet on the membrane. This would be a particularly powerful mechanism if rafts could move directionally or if raft reorganization were to take place under the influence of some signal. Additionally, increasing the collision rate (as observed in our model) would also serve to increase the efficiency of membrane reactions. An interesting observation from the modeling was that the accentuated collision rate was realized for only small rafts with diameters in the range of 6 to 14 nm. There have been many different estimates of raft size that reflect the different methodologies used to study this type of membrane microdomain. The most recent estimates using fluorescence resonance energy transfer in live cells and EM in fixed cells suggests that the size of cholesterol-rich microdomains may be much smaller than originally suggested and have proposed diameters in the range of 5 to 20 nm (20, 22, 24, 31). Our study here suggests that microdomains of this size are optimal for promoting intermolecular collisions between captured proteins. This conclusion is further supported by recent estimates of the diameters of nonraft H-, N-, and K-Ras microdomains that fall in the same range (8, 22). In this context, it is also interesting to consider how long a protein spends in a raft after entering. This is the subject of ongoing work, but for ρ equaling 0.5, we find that the average residence time for a protein in a 6-nm raft is ~25 μs and that for a 14-nm raft is ~60 μs (D. V. Nicolau, Jr., K. Burrage, and J. F. Hancock, unpublished data). Our model therefore suggests that the temporal resolution needed to visualize small lipid rafts by SPT may require an image every 5 to 10 μs, whereas the highest resolution achieved to date is an image every 25 μs used to track gold-labeled GPI-anchored proteins (11, 12, 17).

A further important characteristic of the system is observed if rafts selectively capture proteins, implemented here using reflection probabilities to simulate changes in raft or microdomain affinity. This is a biologically meaningful way to model the interaction, since proteins can have more than one raft or microdomain anchor, resulting in different affinities for rafts and nonraft regions and, accordingly, different likelihoods of diffusing out of rafts (for example, see references 25 and 26). When proteins of specific types are confined in this way to microdomains, selective collisions between proteins of these types can be induced. Remarkably, this can be achieved even if the concentrations of the proteins of interest are low. The amplification of selective collisions is most dramatic at low raft areas. This is an interesting result because it supports the idea that, under certain biophysical constraints, rafts could easily operate as selective concentrators of proteins (present in modest concentrations on the membrane) that need to interact. Note that, although we refer to rafts here, the same conclusion is relevant to any protein that can be trapped in a specific microdomain by some structural change or modification that increases affinity for that domain. These results demonstrate that the generation of low-abundance microdomains with high affinity for activated GTP-loaded Ras proteins would be sufficient to aggregate Ras-GTP into the signaling nanoclusters recently observed by EM and SPT (8, 16).

Taken together, our results therefore suggest that two mechanisms may operate on the plasma membrane to control selective biomolecular interactions in raft domains. Increases in raft or microdomain affinity (prn) promote selective collection of proteins to subsets of rafts for which they have high affinity. This mechanism offers the most flexibility for regulating protein-protein interactions. In the absence of these specific interactions, differences in diffusion rates between raft and nonraft membrane (ρ) presumably mediated by different membrane anchors is an efficient constitutive mechanism to segregate protein sets. And, assuming rafts are small (≤14 nm) and mobile, this diffusion-driven segregation will also facilitate intermolecular collisions. It will be the subject of future work to elucidate the overall effect of rafts on membrane dynamics by including “triggering” mechanisms and multiraft “platform” formation that would simulate in vivo signaling events and organizational phenomena, respectively. Additionally, concurrent work is addressing the question of whether rafts can account (in part) for the anomalous diffusion behavior observed on cell membranes. Finally, it is important to reiterate that our work describes the behavior of modeled proteins and modeled rafts and, therefore the results are dependent on the assumptions of the model.

In conclusion, we have presented a stochastic model of lipid rafts and microdomains. Simulations of this model show that, on short time scales, the mobility of proteins is relatively insensitive to the presence of rafts, while on long time and distance scales, rafts significantly slow the exchange of proteins between membrane regions. Rafts capture proteins if their diffusion is attenuated in rafts, and rafts are capable of capturing specific proteins for which they have high affinity. Specific collision rates between proteins can be greatly increased by their concentration in mobile rafts, but this effect is constrained by raft size and the extent of diffusion attenuation. Finally, it should be noted that this approach and these general conclusions are equally applicable to protein interactions with other types of membrane microdomains. In this context, the idea that the diffusive behavior of proteins is affected by both positive microdomain interaction and exclusion from other microdomains needs further exploration.

Supplementary Material

[Supplemental material]

Acknowledgments

This work was supported by grants from the National Institutes of Health, Bethesda, Md. (GM066717) and the National Health and Medical Research Council, Australia. The IMB is a Special Research Centre of the Australian Research Council. K.B. gratefully acknowledges support via the Federation Fellowship of the Australian Research Council.

We thank Gerardin Solana for the use of computer resources for some of the simulations, Yoav Henis for advice on FRAP calculations, and Alpha Yap for comments on the manuscript.

Footnotes

Supplemental material for this article may be found at http://mcb.asm.org/.

REFERENCES

1. Anderson, R. G., and K. Jacobson. 2002. A role for lipid shells in targeting proteins to caveolae, rafts, and other lipid domains. Science 296:1821-1825. [PubMed]
2. Axelrod, D., D. E. Koppel, J. Schlessinger, E. L. Elson, and W. W. Webb. 1976. Mobility measurement by analysis of fluorescence photobleaching recovery kinetics. Biophys. J. 16:1055-1069. [PMC free article] [PubMed]
3. Berry, H. 2002. Monte Carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation. Biophys. J. 83:1891-1901. [PMC free article] [PubMed]
4. Dietrich, C., B. Yang, T. Fujiwara, A. Kusumi, and K. Jacobson. 2002. Relationship of lipid rafts to transient confinement zones detected by single particle tracking. Biophys. J. 82:274-284. [PMC free article] [PubMed]
5. Edidin, M. 2003. The state of lipid rafts: from model membranes to cells. Annu. Rev. Biophys. Biomol. Struct. 32:257-283. [PubMed]
6. Elad-Sfadia, G., R. Haklai, E. Balan, and Y. Kloog. 2004. Galectin-3 augments K-Ras activation and triggers a Ras signal that attenuates ERK but not phosphoinositide 3-kinase activity. J. Biol. Chem. 279:34922-34930. [PubMed]
7. Fujiwara, T., K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi. 2002. Phospholipids undergo hop diffusion in compartmentalized cell membrane. J. Cell Biol. 157:1071-1081. [PMC free article] [PubMed]
8. Hancock, J. F., and R. G. Parton. 2005. Ras plasma membrane signalling platforms. Biochem. J. 389:1-11. [PMC free article] [PubMed]
9. Henis, Y. I. 1993. Lateral and rotational diffusion in biological membranes. In M. Shinitzky (ed.), Biomembranes—physical aspects. VCH, Weinheim, Germany.
10. Kenworthy, A. K., B. J. Nichols, C. L. Remmert, G. M. Hendrix, M. Kumar, J. Zimmerberg, and J. Lippincott-Schwartz. 2004. Dynamics of putative raft-associated proteins at the cell surface. J. Cell Biol. 165:735-746. [PMC free article] [PubMed]
11. Kusumi, A., I. Koyama-Honda, and K. Suzuki. 2004. Molecular dynamics and interactions for creation of stimulation-induced stabilized rafts from small unstable steady-state rafts. Traffic 5:213-230. [PubMed]
12. Kusumi, A., C. Nakada, K. Ritchie, K. Murase, K. Suzuki, H. Murakoshi, R. S. Kasai, J. Kondo, and T. Fujiwara. 2005. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: high-speed single-molecule tracking of membrane molecules. Annu. Rev. Biophys. Biomol. Struct. 34:351-378. [PubMed]
13. Kwik, J., S. Boyle, D. Fooksman, L. Margolis, M. P. Sheetz, and M. Edidin. 2003. Membrane cholesterol, lateral mobility, and the phosphatidylinositol 4,5-bisphosphate-dependent organization of cell actin. Proc. Natl. Acad. Sci. USA 100:13964-13969. [PMC free article] [PubMed]
14. Metzler, R., and J. Klafter. 2000. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339:1-77.
15. Munro, S. 2003. Lipid rafts. Elusive or illusive? Cell. 115:377-388 [PubMed]
16. Murakoshi, H., R. Iino, T. Kobayashi, T. Fujiwara, C. Ohshima, A. Yoshimura, and A. Kusumi. 2004. Single-molecule imaging analysis of Ras activation in living cells. Proc. Natl. Acad. Sci. USA 101:7317-7322. [PMC free article] [PubMed]
17. Murase, K., T. Fujiwara, Y. Umemura, K. Suzuki, R. Iino, H. Yamashita, M. Saito, H. Murakoshi, K. Ritchie, and A. Kusumi. 2004. Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques. Biophys. J. 86:4075-4093. [PMC free article] [PubMed]
18. Niv, H., O. Gutman, Y. Kloog, and Y. I. Henis. 2002. Activated K-Ras and H-Ras display different interactions with saturable nonraft sites at the surface of live cells. J. Cell Biol. 157:865-872. [PMC free article] [PubMed]
19. Parton, R. G. 2003. Caveolae—from ultrastructure to molecular mechanisms. Nat. Rev. Mol. Cell Biol. 4:162-167. [PubMed]
20. Parton, R. G., and J. F. Hancock. 2004. Lipid rafts and plasma membrane microorganization: insights from Ras. Trends Cell Biol. 14:141-147. [PubMed]
21. Paz, A., R. Haklai, G. Elad-Sfadia, E. Ballan, and Y. Kloog. 2001. Galectin-1 binds oncogenic H-Ras to mediate Ras membrane anchorage and cell transformation. Oncogene 20:7486-7493. [PubMed]
22. Plowman, S., C. Muncke, R. G. Parton, and J. F. Hancock. 2005. H-ras, K-ras and inner plasma membrane raft proteins operate in nanoclusters that exhibit differential dependence on the actin cytoskeleton. Proc. Natl. Acad. Sci. USA 102:15500-15505. [PMC free article] [PubMed]
23. Pralle, A., P. Keller, E. L. Florin, K. Simons, and J. K. Horber. 2000. Sphingolipid-cholesterol rafts diffuse as small entities in the plasma membrane of mammalian cells. J. Cell Biol. 148:997-1008. [PMC free article] [PubMed]
24. Prior, I. A., C. Muncke, R. G. Parton, and J. F. Hancock. 2003. Direct visualization of Ras proteins in spatially distinct cell surface microdomains. J. Cell Biol. 160:165-170. [PMC free article] [PubMed]
25. Rotblat, B., I. A. Prior, C. Muncke, R. G. Parton, Y. Kloog, Y. I. Henis, and J. F. Hancock. 2004. Three separable domains regulate GTP-dependent association of H-ras with the plasma membrane. Mol. Cell. Biol. 24:6799-6810. [PMC free article] [PubMed]
26. Roy, S., S. Plowman, B. Rotblat, I. A. Prior, C. Muncke, S. Grainger, R. G. Parton, Y. I. Henis, Y. Kloog, and J. F. Hancock. 2005. Individual palmitoyl residues serve distinct roles in H-ras trafficking, microlocalization and signaling. Mol. Cell. Biol. 25:6722-6733. [PMC free article] [PubMed]
27. Saffman, P. G., and M. Delbruck. 1975. Brownian motion in biological membranes. Proc. Natl. Acad. Sci. USA 72:3111-3113. [PMC free article] [PubMed]
28. Saxton, M. J. 1994. Anomalous diffusion due to obstacles: a Monte Carlo study. Biophys. J. 66:394-401. [PMC free article] [PubMed]
29. Saxton, M. J. 2001. Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study. Biophys. J. 81:2226-2240. [PMC free article] [PubMed]
30. Schutz, G. J., G. Kada, V. P. Pastushenko, and H. Schindler. 2000. Properties of lipid microdomains visualized by single molecule microscopy. EMBO J. 19:892-901. [PMC free article] [PubMed]
31. Sharma, P., R. Varma, R. C. Sarasij, Ira, K. Gousset, G. Krishnamoorthy, M. Rao, and S. Mayor. 2004. Nanoscale organization of multiple GPI-anchored proteins in living cell membranes. Cell 116:577-589. [PubMed]
32. Simons, K., and E. Ikonen. 1997. Functional rafts in cell membranes. Nature 387:569-572. [PubMed]
33. Simons, K., and D. Toomre. 2000. Lipid rafts and signal transduction. Nat. Rev. Mol. Cell Biol. 1:31-39. [PubMed]
34. Singer, S. J., and G. L. Nicolson. 1972. The fluid mosaic model of the structure of cell membranes. Science 175:720-731. [PubMed]

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