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

The dynamic lateral segregation of signaling proteins into microdomainsis proposed to facilitate signal transduction, but the constraintson microdomain size, mobility, and diffusion that might realizethis function are undefined. Here we interrogate a stochasticspatial model of the plasma membrane to determine how microdomainsaffect protein dynamics. Taking lipid rafts as representativemicrodomains, we show that reduced protein mobility in raftssegregates dynamically partitioning proteins, but the equilibriumconcentration is largely independent of raft size and mobility.Rafts weakly impede small-scale protein diffusion but more stronglyimpede long-range protein mobility. The long-range mobilityof raft-partitioning and raft-excluded proteins, however, isreduced to a similar extent. Dynamic partitioning into raftsincreases specific interprotein collision rates, but to maximizethis critical, biologically relevant function, rafts must besmall (diameter, 6 to 14 nm) and mobile. Intermolecular collisionscan also be favored by the selective capture and exclusion ofproteins by rafts, although this mechanism is generally lessefficient than simple dynamic partitioning. Generalizing theseresults, we conclude that microdomains can readily operate asprotein concentrators or isolators but there appear to be significantconstraints on size and mobility if microdomains are also requiredto function as reaction chambers that facilitate nanoscale protein-proteininteractions. These results may have significant implicationsfor the many signaling cascades that are scaffolded or assembledin plasma membrane microdomains.

INTRODUCTION

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Introduction

The classical view of the plasma membrane lipid bilayer, asa two-dimensional fluid acting as a neutral solvent for membraneproteins in which all particles diffuse freely (34), has beensubstantially modified in recent years. The plasma membraneis a complex structure that is compartmentalized on multiplelength and time scales. This compartmentalization is drivenby a variety of lipid-lipid, lipid-protein, and actin cytoskeletoninteractions (1, 5, 11, 13, 17). These mechanisms are well illustratedby the plasma membrane interactions of the Ras GTPases thathave 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 theplasma membrane in microdomains or nanoclusters. Activated GTP-loadedH-ras and K-ras dynamically partition into nonoverlapping clustersthat are unaffected by cholesterol depletion but exhibit differentialdependence on the actin cytoskeleton (16, 18, 24, 25). ActivatedGTP-loaded N-ras, in contrast to H-ras, partitions into cholesterol-dependentclusters (8, 16, 26). Clustering of H- and K-ras proteins requiresthe common scaffold protein Sur-8 and the selective scaffoldsgalectin-1 and galectin-3, respectively (6, 8, 16, 21). Thereis good evidence that these various Ras microdomains are platformsfor signal transduction (8, 16, 26).

A further example of membrane microdomains is the lateral segregationof glycosphingolipids and cholesterol into liquid-ordered domains.Phase separation of cholesterol-enriched, liquid-ordered domainsor lipid rafts has been clearly demonstrated in model membranesand also in biological membranes, although the length and timescales on which this phase separation occurs are a matter ofdebate (5, 11, 15, 32). Multiple estimates of the diameter oflipid rafts have been provided using diverse techniques, althoughphotonic force microscopy, fluorescence resonance energy transfer,and EM provide a convergence of estimates of 6 to 50 nm, withthe 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 activatedH-ras and K-ras (22, 24).

An important role that has been ascribed to all plasma membranemicrodomains is that of selectively concentrating proteins tofacilitate the assembly of signaling complexes (33). Many studieshave been qualitatively interpreted in terms of this type ofmicrodomain model. However, no quantitative analysis has beenattempted to explore the basic mechanics of how microdomainsmight drive protein-protein interactions, as demanded in theirrole of supporting the assembly of signaling platforms. Forexample, if microdomains do aggregate proteins, are there anyconstraints on size and dynamics that need to be imposed forthem to achieve this function? If so, are these constraintsrealistic and how do the predictions compare with recent estimatesof microdomain size and dynamics? These are difficult but importantquestions that are especially relevant in the context of theongoing discussions of plasma membrane structure and function.

Since microdomains are larger than single proteins, the diffusionrate of proteins sequestered in microdomains is lower than thosein surrounding membranes (23). However, if proteins enter andleave microdomains, i.e., exhibit dynamic partitioning, theexpected local and global effects on protein diffusion are moredifficult to predict. The purpose of this study is to explorethese basic issues by constructing and interrogating stochasticMonte Carlo models of the dynamics of protein molecules on amembrane in the presence of microdomains. For the purpose ofthis paper, we will consider lipid rafts as a specific exampleof a plasma membrane microdomain, given the extensive literaturethat evokes these structures to explain biological observationsand the desire to provide an appropriate computational approachfor evaluating these claims.

MATERIALS AND METHODS

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Materials and Methods

Model description and Monte Carlo simulations. The model relies on the representation of the cell membraneas a two dimensional lattice. Each element of this lattice isa "voxel" that can be either occupied or unoccupied by a modeledprotein at each time step; in the former case, a record is madeof which modeled protein occupies the voxel. At any time, onlyone modeled protein may occupy a given voxel, in order to ensurevolume exclusion between modeled proteins. For brevity, we willsubsequently refer to "modeled proteins" simply as "proteins."Throughout this work, the size of a voxel has been assumed tobe 2 nm by 2 nm, this being an estimate of the average sizeof 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 x 10⁶ voxels),while in all other simulations it is 500 nm by 736 nm (9.2 x10⁴ voxels). The two main considerations in choosing the voxelsize are (i) the size of a membrane-anchored protein (so thatvolume exclusion can be accurate) and (ii) the dimensions ofthe membrane (must be large enough to get accurate statisticsbut small enough to make the simulations tractable).

A lipid raft is modeled as a two-dimensional patch whose boundariesare described by a simple closed curve in the plane. In thiswork, rafts are represented as disks and so a modeled raft occupiesan area in the plane A = ${pi}$ r² and is surrounded by nonraft regionsas well as other modeled rafts. A given voxel is thus assignedto either a raft or a nonraft region. In this study, modeledrafts 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" simplyas "rafts."

The lattice is seeded with proteins of different species [foreach species i, let the number of proteins present in the systeminitially be n_i(0)]. Each protein has two properties: a positionspecified in terms of its x and y coordinates in the latticeand a species. At each step, a protein M is chosen at randomfrom the general population. Let the coordinates of this proteinbe (x,y). One of the voxels with coordinates (x + D_i,y), (x– D_i,y), (x,y + D_i), or (x,y – D_i) is also chosenat random, where D_i (a positive real number) is the step sizeof species i that depends on its location (inside or outsidea raft). The new voxel is the location to which Brownian motionmoves the protein during the current time step alone. Note thatthe case where D equals 1 corresponds to choosing an adjacentvoxel. If the new voxel is occupied by a protein or a fixedobstacle, then protein M is placed back in its original voxel(x,y) and a collision is recorded. This is an implementationof volume exclusion, so that only one protein can occupy a voxelat any time, as described above (3). After each such step, thesimulation time is incremented by 1/n, where n is the totalnumber of proteins on the membrane. Thus, a simulation intervalof 1 corresponds (statistically) to the time needed for allproteins to move once.

The "baseline" step size of a protein in a nonraft region isset to D equals 1, and because rafts move more slowly than proteins,the step size of a raft is less than unity. These values areestimated from the work of Saffman and Delbruck (27) and areshown in Table 1. Since the mobility of a protein in a raftis reduced compared with that of the surrounding membrane region(4, 23, 30), an important parameter that has been used throughoutthis study to describe the interaction of a protein with a raftis the ratio of the step size of a species inside and outsidea model raft ${rho}$ _i.

Formula 1 (1)
Largervalues of D correspond to higher diffusion rates, that is, abetter-mixed system. If D is 0, then the raft or molecular speciesin question is immobile. If D is nonintegral, then the interpretationof D is probabilistic and the size of the diffusive step isnondeterministic. For example, if D_i is 0.5, then a proteinof species i has, at each step, a probability of 0.5 of movingto one of its neighboring voxels (if unoccupied) and an equalprobability of not moving at all. This is used to implementstatistically subvoxel step sizes (the unitary step size mustalways be the size of one voxel, 2 nm). Throughout this study,we have used a step size of 1 voxel outside rafts and a stepsize of ${rho}$ inside rafts. Because the membrane size is large relativeto a voxel size, the algorithm is a good random walk approximationto Brownian motion.

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TABLE 1. Step sizes of rafts relative to proteins^a

The differing numbers of rafts and proteins in the simulationcomplicate the modeling somewhat. If a raft is chosen at randomduring each Monte Carlo step and moved to a new position, thediffusion coefficients of rafts and proteins cannot be equatednumerically. As an example, consider a situation where thereare 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 periodsince it has a 1/10 probability of being chosen during eachof the 100 steps. This is not a physically accurate realization:each raft should only have moved once, to keep diffusion ratesindependent of time. To achieve this, at each Monte Carlo step,the motion of rafts is governed by two random numbers. The firstof these, R₁, is sampled from the uniform distribution on [0,1] and is used to decide whether or not any raft will move duringthis step. Specifically, if R₁ is smaller than the raft-to-proteinratio R_r/m defined by n_rafts/n_proteins, where n_rafts and n_proteinsare the total numbers of rafts and proteins present, respectively,then one raft will move during this step; otherwise, all raftsretain their positions. If it is decided to move a raft, a secondinteger, R₂, is chosen at random between 1 ... n_r_afts and usedto decide which raft will be moved. That raft will be movedin one of the cardinal directions, as for proteins and accordingto its step size. If, however, the new area to be occupied bythe raft overlaps with that occupied by another raft, then itis returned to its old location. Thus, a volume exclusion conditionis imposed between rafts. The parameters tracked during simulationsare (i) the proportion of proteins in rafts, (ii) the equilibriumconcentration of proteins in rafts after allowing the systemto reach a steady state, and (iii) the collision rate of proteins.A collision between two proteins is recorded whenever one ofthe two attempts to move to a voxel that is occupied by theother (and is rejected). The time derivative of the total collisioncount, estimated using a difference formula, is the approximatenumber of collisions per unit time at each point in the simulation.

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

To simulate raft affinity, "rejection probabilities" were introducedassociated with proteins entering and exiting a raft. When aprotein moves from a nonraft voxel to a raft voxel, it may bereturned to its original location (rejected) with probabilityp_rr. Conversely, when exiting a raft, it may be rejected withprobability p_rn. If these probabilities are 0, the proteinsdo not differentiate between rafts and nonraft regions, exceptfor the difference in diffusion rates they experience (i.e.,raft partitioning is always influenced by ${rho}$ irrespective of thevalue of p_rn). At the other extreme, a probability of p_rn orp_rr equal to 1 indicates that once a protein has entered a raftor nonraft region, respectively, it will be permanently capturedin that region.

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

Formula 2 (2)
where D is the diffusion coefficient, ${Gamma}$ (k)is the gamma function defined as

Formula 3 (3)
when k is >0, and ${alpha}$ is the anomalous exponent,a measure of the nonclassical behavior of the Brownian motionexecuted by a particle (a protein in this case). Thus, the coefficientof diffusion can be estimated from the vertical intercept yof a straight line fitting the log(<X(t)²>)-log(t) data

Formula 4 (4)
and ${alpha}$ is estimated from the slopeof the same line. Note that in the case of pure diffusion, ${alpha}$ = 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 valuesof 1. When the system has reached equilibrium with respect tospatial distributions of proteins, all proteins in a particulararea of the membrane have their tags set to 0 (while all proteinsoutside this area have unchanged tags). Subsequently, the totalsum of the tags over the "bleached" area is recorded periodicallyand this procedure is repeated until this sum, representingthe total fluorescence due to the bleached area, returns toits initial value. From this data, t_0.5, the time required forthe fluorescence signal to return to half of its initial valueis extracted (2, 9). The relationship between the diffusioncoefficient and t_0.5 is

Formula 5 (5)
whereD_macro is the large-scale diffusion rate, ${omega}$ is the bleach radius,and ${gamma}$ is a correction factor (0.88 for a circular bleached area).The "macro" subscript refers to the fact that because of therelatively long time scales involved in FRAP experiments (andour simulations), the diffusion coefficient estimated usingthis method reflects the large-scale mobility of proteins. Inthe case of pure diffusion, one would expect the diffusion coefficientto be independent of the time and space scales (it is a constantin the diffusion equation). Recent studies, however, have suggestedthat diffusion is strongly impeded over large space scales sothat the short-range diffusion coefficient D_micro is not, infact, equal to the large-scale coefficient D_macro (28, 29).We estimated both D_micro and D_macro using direct short timescale and FRAP simulations, respectively, in an effort to investigatewhether the presence of rafts accounts for the nonconstancyof the diffusion coefficient.

RESULTS

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Results

Constructing a Monte Carlo model of the plasma membrane. Here we establish a quantitative framework for examining howlipid rafts influence plasma membrane protein dynamics. We considerrafts in the context of their classical representation as domainsenriched in cholesterol and sphingolipids that diffuse as stableentities within the fluid bilayer (23, 33). Although we referspecifically to lipid rafts, it should be stressed that themodel is generally applicable to any type of plasma membranemicrodomain that diffuses as a stable entity within the fluidbilayer, irrespective of whether the integrity of the microdomainis 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-disorderedphase; or class 3, dynamic partitioning proteins that move inand out of rafts; although classes 1 and 2 are simply extremerepresentations of class 3. If rafts are mobile, a protein ina raft will be acted on by two independent sources of motion:its own diffusive motion and the diffusion of the raft thatcontains it. If a protein is excluded from rafts, its mobilityand diffusion will also be influenced by the sea of rafts inwhich it operates.

A Monte Carlo approach is particularly suited to the problemsconsidered here because the local motions of proteins and raftscan be implemented using simple transition rules, while individualevents, such as collisions between proteins, can easily be recordedin time. A detailed description of the model is provided inMaterials and Methods. In brief, a model lattice membrane wasconstructed onto which proteins and rafts were randomly seeded.The proteins and rafts were then moved randomly by Brownianmovement. The area of the membrane that was designated lipidraft and the radius of the rafts were varied over wide rangesbut were fixed during any given simulation. In the model, raftsmove more slowly than individual proteins because of their greatercross-sectional diameter. The relative velocity of rafts ofa given radius compared to that of an individual protein wasestimated using the Saffman-Delbruck equation (27) and is shownin Table 1. The mobility of a protein in a raft is reduced comparedwith that of the surrounding membrane region because of thedense packing of lipids in the liquid-ordered raft environment(4, 23, 30). This critically important behavior is reflectedin the parameter ${rho}$ _i, which is the ratio of the distances movedby a protein in unit time, inside and outside a raft (see equation1 in Materials and Methods). The influence of this parameteron the behavior of protein diffusion and protein-protein interactionsforms the first part of this study.

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

To fully characterize the properties of these model rafts, wefirst examined in detail the situation where rafts have no inherentaffinity for a protein but operate simply under the parameter ${rho}$ . These simulations therefore reflect the behavior of proteinsthat interact with the plasma membrane primarily via N- or C-terminallipid anchors. If the diffusion rate of a protein is lower inrafts, then, intuitively, the residence time of a protein ina raft should, on average, be greater than that in a nonraftregion of equal dimensions. In other words, proteins are morelikely to be found inside rafts than outside them if ${rho}$ is <1.This hypothesis was tested with simulations in which the initialdistribution of proteins was uniform and random: the dimensionsof the membrane were 500 nm by 736 nm onto which 2,500 proteinswere placed. Representative results are shown in Fig. 1. Theproportion of proteins in rafts begins at 25% since, in thissimulation, rafts represent 25% of the membrane, and proteinsare distributed randomly. As proteins enter rafts through diffusion,they are less likely to escape back to nonraft regions due tothe reduced rate of diffusion rate in the raft domains. Thisleads to an aggregation of proteins in rafts, which becomesmore pronounced as the value of ${rho}$ decreases. Conversely if ${rho}$ is1, so that proteins are diffusively "blind" to the presenceof rafts, the effect is negligible. If rafts are immobile and ${rho}$ is 1, there is no aggregation. If rafts are mobile and ${rho}$ is1, a minor amount of aggregation is observed due to the "sweepingup" of proteins by mobile rafts; for example there is an 3%increase in Fig. 1. A raft "swallows" proteins that are in itspath, and these are subsequently moved along with the raft duringtheir period of residence. Figure 1 also shows that, after atransient period during which the proteins are aggregating inrafts, an equilibrium concentration is reached at which theincreased residence time in rafts is balanced by the nonzeroprobability of escape.

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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. ${rho}$ is the ratio of diffusion rates inside and outside rafts. In this set of simulations, rafts make up 25% of the plasma membrane and are mobile. Raft diameter is 14 nm. Note there is aggregation of proteins in rafts when there is a reduced diffusion rate in rafts (i.e., when ${rho}$ is <1).

The equilibrium concentration is essentially independent of raft size and mobility. How does the equilibrium concentration vary with raft dimensionsand with ${rho}$ ? To answer this question, we performed experimentsusing all combinations of the following parameters: (a) totalraft area equaling 10%, 25%, and 50% of the membrane, (b) raftdiameters equaling 6 nm, 14 nm, 26 nm, and 50 nm, (c) raftsimmobile and mobile, and (d) ${rho}$ equaling 0.25, 0.5, 0.75, and1. The membrane area and number of proteins are the same asin Fig. 1, and the simulations were run to equilibrium. A completeset of results is shown in Fig. 2. The most striking featureof these results is that the equilibrium concentration is essentiallyindependent of raft diameter and is not greatly affected, overall,by raft mobility. The equilibrium concentration of proteinsin rafts falls, albeit quite weakly, with increasing ${rho}$ ; as ${rho}$ $->$ 1,this concentration tends to the proportion of the membrane thatthe rafts occupy, as would be expected since, when ${rho}$ equals 1,rafts are either blind (in the immobile case) or almost so (inthe mobile case) to the presence of rafts. Conversely, at lowvalues of ${rho}$ , the probability of escaping from a raft is low andthe aggregation effect is substantial, leading to a high equilibriumconcentration in rafts. Thus, as ${rho}$ $->$ 0, the rafts act increasinglyas "protein sinks." The dependence of equilibrium protein concentrationon ${rho}$ appears moderately stronger as total raft area decreases,but this is so simply because the initial concentration in raftsthen also decreases. We conclude, therefore, that the equilibriumconcentration of a protein with no inherent affinity for raftsis predominantly independent of raft dimensions and raft mobilitybut moderately and approximately linearly dependent on the differencein diffusion rates of that protein between raft and nonraftregions.

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FIG. 2. Equilibrium concentrations of proteins in lipid rafts. As in Fig. 1, proteins have no intrinsic affinity for rafts; association is driven by the parameter ${rho}$ . Simulations were run for four values of ${rho}$ , with rafts occupying 10 to 50% of the plasma membrane with diameters of 6 to 50 nm. The equilibrium concentration of proteins in lipid rafts depends modestly on ${rho}$ in both cases and weakly on raft diameter.

Rafts have modest effects on (small-scale) diffusion rates of proteins. We next investigated to what extent the average diffusion rateof proteins is influenced by the presence of rafts. The diffusioncoefficient was estimated using equation 2 from the mean squareddisplacement data recorded during the simulations. Figure 3shows that the diffusion coefficient grows with ${rho}$ , and the dependenceis approximately linear for immobile rafts and is not strongbut increases somewhat with raft area. If rafts are mobile,the dependence is more tenuous and reveals a complex interplaybetween raft size (and hence diffusion rate), ${rho}$ , and total raftarea. In the immobile case, the diffusion coefficients convergeas ${rho}$ $->$ 1, while in the mobile case they do not, due to differencesbetween the mobility of differently sized rafts (Table 1). Thediffusion coefficients in the presence of mobile rafts are alwaysgreater than when rafts are immobile, but the separation issmall and only fully resolved when the raft area is $≥$ 25%.

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FIG. 3. The coefficient of diffusion of proteins in the presence of lipid rafts. As in Fig. 1, proteins have no intrinsic affinity for rafts; association is driven by the parameter ${rho}$ . Simulations were run for four values of ${rho}$ , with rafts occupying 10 to 50% of the plasma membrane with diameters of 6 to 50 nm. D is estimated from mean squared deviation data using equation 2.

Figure 3 shows that the plots of D versus ${rho}$ for immobile raftsof different diameters are superimposed on each other. Thus,if rafts are immobile, raft diameter has no effect on the diffusioncoefficient. In contrast, the plots of D versus ${rho}$ for mobilerafts do show some weak separation at high raft areas with 6-nmrafts slowing diffusion when ${rho}$ equals 0.25 to a greater extentthan larger rafts. We conclude that rafts have only a moderateeffect on the diffusion coefficients of raft-partitioning proteins.In the most extreme realization of the raft model, where 50-nmdiameter immobile rafts occupying 50% of the cell surface and ${rho}$ equals 0.25 so that 90% of the proteins are partitioned inthe rafts at any given time point (Fig. 2), diffusion wouldbe slowed only $~$ 2.5-fold compared to the same protein diffusingin a plasma membrane containing no rafts (Fig. 3). For a singleprotein diffusing on a free membrane, the maximum value of Dis exactly 0.5. Note, however that in some simulations withmobile rafts, D is higher than this value. This phenomenon isperhaps due to the combination of the two independent movementsof a protein: (i) Brownian motion and (ii) the movement of theraft in which the protein is embedded. These two effects maybe roughly additive, resulting in the observed diffusion rateof 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 occurringinside and outside of rafts in all of the simulations summarizedin Fig. 2 and 3 and calculated a collision rate as describedin Materials and Methods.These results show that if rafts areimmobile, the collision rate is insensitive to both raft sizeand the parameter ${rho}$ (Fig. 4). In a sense, this is not unexpected.Although the concentration of proteins in rafts is higher thanin the surrounding membrane, leading potentially to a higherrate of collision, this is offset by the lower diffusion coefficientin rafts causing reduced protein mobility and hence reducingthe rate of collisions. However, this is not the case if raftsare mobile. Simulations show that the collision rate dependson ${rho}$ , reaching a maximum at around ${rho}$ equals 0.5 and falling awayat smaller and larger values. The collision rate also variesstrongly with raft size but in a complex interplay with raftarea. Thus, 6-nm mobile rafts allow the highest collision rateswhen the raft area is 10% but allow the lowest when raft areais 50%. In contrast, a raft diameter of 14 nm maximizes collisionrates at areas >25% and provides close to maximal collisionrates at 10% area. We speculate that the increase in collisionrate observed when rafts are mobile is due to mainly collisionsbetween rafts, which bring their constituent proteins (especiallythose close to the rims of the rafts) in close proximity. Weconclude that if the function of rafts is to facilitate protein-proteininteractions, then raft mobility is critical to realizing thisfunction. Moreover, the model suggests that an optimal diameterfor rafts to maximize collisions between raft-partitioning proteinsis $~$ 14 nm. The maximum collision rate is reached when ${rho}$ equalsapproximately 0.5, which represents a good compromise betweenincreased concentration of proteins in rafts and reduced proteinmobility.

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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. 2 and 3. As in Fig. 1, proteins have no intrinsic affinity for rafts; association is driven by the parameter ${rho}$ . Simulations were run for four values of ${rho}$ , with rafts occupying 10 to 50% of the plasma membrane with diameters of 6 to 50 nm. The total collision rate is the approximate number of collisions per unit time at each point in the simulation. (Note that this includes collisions taking place inside and outside rafts.)

Picket-fence experiments. Given that the data presented earlier suggests that the effectof rafts on protein mobility on the small scale (D_micro) isnot 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 raftdimensions were set to 14 nm, ${rho}$ to 0.5, with a raft coveragearea of 50%, and simulations were run for 500 time steps. Theeffect of the fence on mobile and immobile rafts was testedbecause, as described in Materials and Methods, collisions withthe fence impede the movement of mobile rafts. The results areshown in Table 2 and compared with earlier data in which therewas no fence post arrangement. We find that the presence ofthe fence has an insignificant effect on the collision rateof proteins if rafts are fixed but moderately decreases theinterprotein collision rate if rafts are mobile. The diffusionrate increases very slightly in the presence of a fence system,and the equilibrium proportions of proteins in rafts fall byabout 10%. We conclude that the presence of a picket fence systemcharacterized by the parameters above has a relatively minoreffect on the dynamics of the system.

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TABLE 2. Effects of an actin fence^a

In silico FRAP experiments. FRAP is frequently used to estimate the lateral diffusion ofplasma membrane proteins over large distance and time scales.A commonly held and perhaps intuitive expectation is that thelateral diffusion rate of raft-associated proteins is significantlylower than nonraft proteins because rafts diffuse slower thansingle proteins. The analysis presented in Fig. 3 suggests,however, that the diffusion coefficients might, in fact, beexpected to vary by only <2.5-fold even in a most extremerealization of the raft model. To compare the actual diffusionrates measured in Fig. 3 with the results of a FRAP experiment,we simulated FRAP by first allowing the distribution of proteinson the membrane to reach steady state over 500 simulation timeunits and then "photobleaching" a circle with a diameter of500 nm. The total area of the membrane was increased to 5 µmby 7.36 µm, so that the number of bleached proteins was<0.6% of the total number of proteins. This ensures thatthe 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 membranewith ( ${rho}$ = 0.5), (iii) mobile rafts of 14 nm diameter making up25% of the membrane with ( ${rho}$ = 0.5), and (iv) mobile rafts of14 nm diameter making up 25% of the membrane but with proteinsexcluded from rafts (i.e., if a protein attempts to enter araft region, it is rejected and does not move from its currentvoxel). Figure 5 shows that the fluorescence signal recoversmuch more slowly if rafts are present. For example, the half-recoverytime is around 1,950 time steps if no rafts are present, whileif rafts represent 50% of the total membrane area, half-recoverytime is around 25,500 time steps. Calculation of the large-scalediffusion coefficients (D_macro) from these data (Table 3) showsthat these differences correspond to an approximately 3.2-foldincrease in D_macro if rafts are absent. Thus, although the localdiffusion rate of proteins is not substantially altered underthese two realizations of the model (a 1.2-fold increase ifrafts are absent), their long-range mobility is significantlyimpeded by the presence of rafts. Moreover, Table 3 also showsthat decreasing raft area from 50% to 25% results in only a1.2-fold increase in D_macro; thus, a substantial change in thearea of the membrane designated raft translates into a relativelymodest change in D_macro. We next examined if a simple FRAP experimentcould discriminate between a raft-associated and raft-excludedprotein under a realization of the raft model where 25% of themembrane is raft. Figure 5 and Table 3 show the interestingresult that exclusion from rafts results in only a minimal,1.07-fold faster D_macro than partitioning into rafts. A comparisonof the calculated D_macro from the FRAP data with the measuredD_micro from the simulations shows that, whereas the two coefficientstrend in the same direction, their relationship to each other,both in the presence and absence of rafts, is highly variable(Table 3). To further explore the reasons for the differentratios predicted by the measured diffusion coefficients andthe estimated values from the FRAP experiments, we observedthe recovery directly in QuickTime movies. For two movies showingrecovery of fluorescence for (a) no rafts and (b) 50% raftswith ${rho}$ equaling 0.5, see the supplemental material. These indicateclearly the difficulty of fluorescent proteins in entering thebleached area and that of bleached proteins in exiting it, respectively,when rafts are present.

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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 ( ${rho}$ = 0.5); or rafts cover 25% of the total area, have 14-nm diameters and slow down diffusion by a factor of 0.5 ( ${rho}$ = 0.5); or rafts represent 25% of the membrane and proteins are excluded from rafts (p_rr = 1). Note the bleaching step at 500 time steps. The t_0.5 for recovery was extracted from the recovery curves (2, 9). See the supplemental material for QuickTime movies of two of these experiments.

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TABLE 3. Long range diffusion is impeded by rafts^a

Can raft affinity facilitate protein-protein interactions? To date, we have considered ${rho}$ as the sole parameter for drivingraft association. We next investigated how an alternative mechanismof confining proteins to raft (or nonraft) regions affects thedynamics of the population. To this end, we distributed 1,000proteins on the plasma membrane and compared the number of collisionswhen a protein is unable to escape from rafts once it has entered(p_rn = 1) with the number of collisions when there is no imposedconfinement (p_rn = 0). The value of the parameter p_rn givesthe probability that a protein attempting to leave a raft willbe rejected; it is a simple realization of raft affinity. Ifraft affinity is high, then p_rn is 1 and, if raft affinity islow, then p_rn is 0 (for details, see Materials and Methods).Since p_rn is a separate parameter from ${rho}$ , we evaluated the effectof raft affinity for two values of ${rho}$ .

First, to examine the effect of p_rn in isolation, ${rho}$ was set to1 so that the diffusion rate in rafts is the same as outside.The simulations were run with 14-nm mobile rafts occupying 10to 50% of the plasma membrane. Figure 6 shows that increasingraft affinity from p_rn equals 0 to p_rn equals 1 substantiallyincreases collision rate. The amplification is greatest (approximatelyfivefold) when raft area is smallest (10%). Second, we repeatedthe same set of simulations but with a diffusion rate reductionin rafts also operating on the proteins (i.e., ${rho}$ was set to 0.5).Interestingly, Fig. 6 shows that, under these conditions, thereis no significant increase in collision rate by increasing raftaffinity, except again when raft area is low. Taking these resultstogether, we conclude that increasing raft affinity (p_rn) maybe an efficient mechanism for concentrating proteins but, interms of promoting protein-protein interactions, raft affinityis not necessarily additive with lateral segregation promotedby diffusion retardation ( ${rho}$ ), unless the area occupied by raftsis small ( $≤$ 10%).

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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 (p_rn) set to 0 or 1 (the minimum and maximum values). The simulations were run for multiple raft areas. When ${rho}$ equals 1 (i.e., the local diffusion rate inside rafts is the same as in nonraft regions), increasing raft affinity produces a significant increase in collision rate (top panel and table), an effect that becomes more pronounced as raft area decreases. In contrast, the effect of changing p_rn from 0 to 1 is much less dramatic if ${rho}$ is already optimal for promoting collisions (i.e., ${rho}$ = 0.5; middle panel and table), although an effect is still seen when the raft area is low (10%). The amplification of collision rate is the ratio of collision rates when p_rn = 1 and p_rn = 0.

DISCUSSION

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Discussion

In this study, we have devised and interrogated a stochasticquantitative model of microdomains, such as those exemplifiedby lipid rafts. We have focused on two fundamental aspects ofprotein dynamics on the plasma membrane that intuitively shouldbe influenced by the presence of lipid rafts: protein mobilityor diffusion and protein collision rates. We have not formallyexamined the effect of lipid rafts on the velocity of specificbiochemical processes, but the analysis of collision rates isa first step in this direction.

The effects of rafts on protein diffusion. The results presented in Fig. 1 to 3 suggest that the mobilityof proteins is not affected to a great degree by the presenceof rafts. On the one hand, this is somewhat surprising sincethe membrane is divided into regions of quite different diffusioncharacteristics (especially as ${rho}$ $->$ 0). On the other hand, if weconsider the membrane to be a homogenous medium, a proportionp_r of which is raft, with ${rho}$ _r being the step size ratio in rafts,then we obtain an estimate for the effective relative diffusionrate ( ${rho}$ _eff) over the membrane, namely,

Formula 6 (6)
so that ${rho}$ _eff varies linearly with the diffusionratio. Of course, the approximation of homogeneity is debatablebecause the rafts are not infinitesimally small. However, asthey are small relative to the membrane, it appears that equation6 is an accurate approximation. In essence, these results indicatethat, within the framework of this microdomain model, no significantnonclassical behavior is caused by the presence of rafts onthe small spatial scale. Rather, the main effect is to attenuatethe global diffusion rate of proteins. Further evidence forthis comes from the anomalous exponents computed using the meansquared deviation data (see equations 2 to 4 in Materials andMethods). The case where ${alpha}$ equals 1 corresponds to classicaldiffusion, while ${alpha}$ < 1 corresponds to anomalous diffusion,in which the mean squared deviation exhibits a power law relationshipin time. In the simulations, the mean value of ${alpha}$ over all simulationswas found to be 0.973 for both the mobile and immobile raftcases. Minimum values of ${alpha}$ were obtained when 50% of the membranewas covered with 6-nm rafts and ${rho}$ was 0.25 and when ${alpha}$ was 0.86or 0.85 if the rafts were mobile or immobile, respectively.Values of ${alpha}$ in the range 0.8 to 0.9 indicate a small to moderatedeparture from classical behavior. In any case, ${rho}$ = 0.25 is atthe low end of the biologically meaningful range of diffusionattenuation factors (estimates of ${rho}$ in the range 0.33 to 0.5have been reported) (4, 23). If ${rho}$ is increased to 0.5, the correspondingminimum values of ${alpha}$ increase to 0.94 and 0.92, respectively.

Given these results, how are we to explain the FRAP data ofFig. 5, showing a very large difference in recovery speeds whenrafts are present or absent? The diffusion rate data suggeststhat the mobility of proteins is not greatly altered by raftson the short time scale. On long time scales, however, the collisionsof rafts with one another reduces their mobility and, sincea large proportion of proteins are in rafts, this in turn mayprevent proteins from exiting the bleached area. Nonbleachedproteins entering the area are likely to collide with proteinsalready occupying this area, resulting in a low rate of exchangebetween the bleached and nonbleached regions. These effectscan be readily appreciated in the movies in the supplementalmaterial. The results in Table 3 also show that relating D_macroto raft area is problematic; decreasing raft area from 50% to25% results in only a 1.2-fold increase in D_macro, whereas decreasingraft area from 25% to 0% results in a 2.5-fold increase in D_macro.In the context of biological experiments that use techniquessuch as cholesterol-depletion to dissemble lipid rafts, thesecomparisons become important since they show that quite minorchanges in D_macro could reflect substantial changes in raftarea. The data also illustrate the difficulty of using FRAPto discriminate between raft-associated and nonraft-associatedproteins. Where 25% of the membrane is raft, exclusion fromrafts results in only a 1.07-fold larger D_macro than partitioninginto rafts, again a very small difference. Thus, the seeminglyintuitive expectation that because a protein is in a raft itshould have a diffusion rate that is significantly lower thana raft-excluded protein is not necessarily realized. These resultsmay explain the similar range of D_macro values reported forraft- and nonraft-partitioning proteins in a recent comprehensivestudy (10). Finally the data illustrate that D_macro deviatessignificantly from D_micro. This has been commented on beforein various studies on membrane protein diffusion (28, 29), butwe show here that this deviation is variable and unpredictablein the presence of lipid rafts.

Interestingly, the presence of an actin fence of low densityreduces the equilibrium concentration of proteins in rafts.The reason for this is that, relative to rafts, proteins areconsiderably more mobile in the presence of even a low-densityfence. While rafts are practically totally impeded from crossinginto a different membrane compartment, proteins can often slipthrough the fence. The cytoskeletal fence system does not, however,result in changed overall protein mobility, ruling out, in theframework of this model, notable nonclassical behavior due tosuch a structure. The fence arrangement at best slightly increasesthe diffusion rate of proteins (despite the fact that it providesadditional obstacles to movement). The picket-fence arrangement,in our model, has only a modest effect on the dynamics of thesystem because even at high fence-post densities, the primaryeffect is to reduce the mobility of rafts. In the limit as raftsbecome completely fixed, the value of D would be reduced byaround only 0.1 to 0.15 (see Fig. 3).

The effects of rafts on protein-protein interactions. An intriguing result from the modeling with significant biologicalimplication is that the collision rate between proteins doesnot behave in the same way in the cases of fixed and mobilerafts, and, furthermore, the dynamics of the system are different.The lateral diffusion rates of rafts are always less than thoseof proteins, albeit modestly as shown in Table 1, so that themotions of proteins might be expected to dominate. Nevertheless,the collision rates when mobile rafts are present are almost1 order of magnitude higher relative to when rafts are fixedor not present (Table 2). We speculate that the reason for theincreased collision rate is that a moving raft, in our model,carries its resident proteins with it. When the raft, throughits own diffusion or through external diffusion, captures oneor more new proteins, this may result in collisions. As proteinsaggregate in rafts, this effect is all the more pronounced.These observations have a clear biological relevance. Caveolaeare large (diameter, 65 nm) immobile raft domains that can occupy4 to 35% of the area plasma membrane according to cell type(19), whereas noncaveolar rafts, at least those occupied byGPI-anchored proteins are smaller (<10 nm) and mobile (31).Segregating proteins in caveolae rather than noncaveolar raftswould, on the basis of our simulations, be predicted to havequite different consequences.

These results suggest a role for rafts as membrane "reactionvessels," increasing the chance that two specific proteins oflow concentration will meet on the membrane. This would be aparticularly powerful mechanism if rafts could move directionallyor if raft reorganization were to take place under the influenceof some signal. Additionally, increasing the collision rate(as observed in our model) would also serve to increase theefficiency of membrane reactions. An interesting observationfrom the modeling was that the accentuated collision rate wasrealized for only small rafts with diameters in the range of6 to 14 nm. There have been many different estimates of raftsize that reflect the different methodologies used to studythis type of membrane microdomain. The most recent estimatesusing fluorescence resonance energy transfer in live cells andEM in fixed cells suggests that the size of cholesterol-richmicrodomains may be much smaller than originally suggested andhave proposed diameters in the range of 5 to 20 nm (20, 22,24, 31). Our study here suggests that microdomains of this sizeare optimal for promoting intermolecular collisions betweencaptured proteins. This conclusion is further supported by recentestimates of the diameters of nonraft H-, N-, and K-Ras microdomainsthat fall in the same range (8, 22). In this context, it isalso interesting to consider how long a protein spends in araft after entering. This is the subject of ongoing work, butfor ${rho}$ equaling 0.5, we find that the average residence time fora protein in a 6-nm raft is $~$ 25 µs and that for a 14-nmraft is $~$ 60 µs (D. V. Nicolau, Jr., K. Burrage, and J.F. Hancock, unpublished data). Our model therefore suggeststhat the temporal resolution needed to visualize small lipidrafts by SPT may require an image every 5 to 10 µs, whereasthe highest resolution achieved to date is an image every 25µs used to track gold-labeled GPI-anchored proteins (11,12, 17).

A further important characteristic of the system is observedif rafts selectively capture proteins, implemented here usingreflection probabilities to simulate changes in raft or microdomainaffinity. This is a biologically meaningful way to model theinteraction, since proteins can have more than one raft or microdomainanchor, resulting in different affinities for rafts and nonraftregions and, accordingly, different likelihoods of diffusingout of rafts (for example, see references 25 and 26). When proteinsof specific types are confined in this way to microdomains,selective collisions between proteins of these types can beinduced. Remarkably, this can be achieved even if the concentrationsof the proteins of interest are low. The amplification of selectivecollisions is most dramatic at low raft areas. This is an interestingresult because it supports the idea that, under certain biophysicalconstraints, rafts could easily operate as selective concentratorsof proteins (present in modest concentrations on the membrane)that need to interact. Note that, although we refer to raftshere, the same conclusion is relevant to any protein that canbe trapped in a specific microdomain by some structural changeor modification that increases affinity for that domain. Theseresults demonstrate that the generation of low-abundance microdomainswith high affinity for activated GTP-loaded Ras proteins wouldbe sufficient to aggregate Ras-GTP into the signaling nanoclustersrecently observed by EM and SPT (8, 16).

Taken together, our results therefore suggest that two mechanismsmay operate on the plasma membrane to control selective biomolecularinteractions in raft domains. Increases in raft or microdomainaffinity (p_rn) promote selective collection of proteins to subsetsof rafts for which they have high affinity. This mechanism offersthe most flexibility for regulating protein-protein interactions.In the absence of these specific interactions, differences indiffusion rates between raft and nonraft membrane ( ${rho}$ ) presumablymediated by different membrane anchors is an efficient constitutivemechanism to segregate protein sets. And, assuming rafts aresmall ( $≤$ 14 nm) and mobile, this diffusion-driven segregationwill also facilitate intermolecular collisions. It will be thesubject of future work to elucidate the overall effect of raftson membrane dynamics by including "triggering" mechanisms andmultiraft "platform" formation that would simulate in vivo signalingevents and organizational phenomena, respectively. Additionally,concurrent work is addressing the question of whether raftscan account (in part) for the anomalous diffusion behavior observedon cell membranes. Finally, it is important to reiterate thatour work describes the behavior of modeled proteins and modeledrafts and, therefore the results are dependent on the assumptionsof the model.

In conclusion, we have presented a stochastic model of lipidrafts and microdomains. Simulations of this model show that,on short time scales, the mobility of proteins is relativelyinsensitive to the presence of rafts, while on long time anddistance scales, rafts significantly slow the exchange of proteinsbetween membrane regions. Rafts capture proteins if their diffusionis attenuated in rafts, and rafts are capable of capturing specificproteins for which they have high affinity. Specific collisionrates between proteins can be greatly increased by their concentrationin mobile rafts, but this effect is constrained by raft sizeand the extent of diffusion attenuation. Finally, it shouldbe noted that this approach and these general conclusions areequally applicable to protein interactions with other typesof membrane microdomains. In this context, the idea that thediffusive behavior of proteins is affected by both positivemicrodomain interaction and exclusion from other microdomainsneeds further exploration.

ACKNOWLEDGMENTS

This work was supported by grants from the National Institutesof Health, Bethesda, Md. (GM066717) and the National Healthand Medical Research Council, Australia. The IMB is a SpecialResearch Centre of the Australian Research Council. K.B. gratefullyacknowledges support via the Federation Fellowship of the AustralianResearch Council.

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

FOOTNOTES

* Corresponding author. Mailing address: Institute for Molecular Bioscience, University of Queensland, Brisbane 4072, Australia. Phone: 61 7 3346 2033. Fax: 61 7 3346 2011. E-mail: j.hancock{at}imb.uq.edu.au.

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

REFERENCES

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