Biophysical mechanism for Ras-nanocluster formation and signaling in plasma membrane
aa r X i v : . [ q - b i o . S C ] J un Biophysical mechanism for Ras-nanocluster formation and signaling in plasmamembrane
Thomas Gurry , , Ozan Kahramano˘gulları , , Robert G. Endres , , ∗ Centre for Integrated Systems Biology at Imperial College,Imperial College London,London SW7 2AZ, United Kingdom, Department of Applied Mathematics and Theoretical Physics,University of Cambridge,Cambridge CB3 0WA, United Kingdom, Department of Computing, Imperial College London,London SW7 2AZ, United Kingdom, Division of Molecular Biosciences,Imperial College London, London SW7 2AZ ∗ To whom correspondence should be addressed. E-mail: [email protected] (Dated: October 29, 2018)Ras GTPases are lipid-anchored G proteins which play a fundamental role in cell signaling pro-cesses. Electron micrographs of immunogold-labeled Ras have shown that membrane-bound Rasmolecules segregate into nanocluster domains. Several models have been developed in attemptsto obtain quantitative descriptions of nanocluster formation, but all have relied on assumptionssuch as a constant, expression-level independent ratio of Ras in clusters to Ras monomers (clus-ter/monomer ratio). However, this assumption is inconsistent with the law of mass action. Here,we present a biophysical model of Ras clustering based on short-range attraction and long-rangerepulsion between Ras molecules in the membrane. To test this model, we performed Monte Carlosimulations and compared statistical clustering properties with experimental data. We find that wecan recover the experimentally-observed clustering across a range of Ras expression levels, withoutassuming a constant cluster/monomer ratio or the existence of lipid rafts. In addition, our modelmakes predictions about the signaling properties of Ras nanoclusters in support of the idea thatRas nanoclusters act as an analog-digital-analog converter for high fidelity signaling.
I. INTRODUCTION
Plasma membrane heterogeneity is a key concept inmolecular cell biology due to its role in protein sortingand specificity of signaling [1, 2, 3]. Although the diver-sity of the membrane’s lipid components is partly respon-sible for this heterogeneity [4], the role played by mem-brane proteins is less well understood. Members of theRas protein superfamily [5, 6] have been observed to formdynamic, non-overlapping domains called nanoclusters inthe inner leaflet of the plasma membrane [7, 8, 9, 10].While the lateral segregation of Ras may provide evi-dence towards the existence of small, dynamic rafts [11],the definition and even existence of rafts remains dis-puted [12]. In addition to its connection to the lipid-raftconcept, Ras has attracted immense interest due to itsfundamental role in a multitude of cellular processes, in-cluding cell proliferation, survival, and motility. Mostimportantly, Ras genes are found to be mutated in 30%of human cancers [13, 14, 15], making their products ex-tremely important therapeutic targets [16]. While the in-tracellular biochemistry of Ras genes is well documented,the biophysical mechanism and role of Ras clustering inthe plasma membrane remains little understood.Ras GTPases are small (21 kDa), lipid-anchored pe-ripheral membrane proteins involved in signal transduc-tion [13]. Three Ras isoforms H-Ras, K-Ras and N-Rasare expressed in all mammalian cells. These isoforms contain a conserved G-domain which binds guanine nu-cleotides [17]. Ras effectively acts as a molecular switchfor the signal, with “on” (GTP-bound) and “off” (GDP-bound) states, the former promoting an association withand activation of effector proteins. Although nearly iden-tical with respect to their catalytic and effector-bindingproperties, H-Ras, N-Ras and K-Ras have very differentbiological roles. This functional distinction is believedto result at least in part from the differential membranecompartmentalization of Ras isoforms [18, 19]. The dif-ferent distribution of Ras proteins in cellular membranesdictates unique spatio-temporal patterns of activationof effector pathways. A classical example of a path-way involving Ras is the Ras-Raf-MEK-ERK pathway,a mitogen-activated protein kinase (MAPK) cascade in-volved in cell proliferation, differentiation, and apoptosis.In this pathway, the epidermal growth factor receptor(EGFR), a receptor tyrosine kinase, is stimulated. Thisleads to recruitment and activation of guanine nucleotideexchange factors (GEFs) which, by interacting with theRas G-domains, promote the exchange of GDP for GTP[17] and lead to Ras activation. Ras · GTP activates pro-tein kinase Raf and initiates the phosphorylation cas-cade, ultimately leading to double phosphorylated ERK(ERKpp) which then travels into the nucleus and phos-phorylates transcription factors [20]. Among other pur-poses, such cascades can lead to a massive amplificationof the original signal [20].
FIG. 1: Experimental immunoEM data and statistical clus-tering analysis. (A) Electron micrograph of immunogold-labeled Ras domain (GFP-tH where tH is minimal plasmamembrane targeting motifs of H-Ras) in an in vitro plasmamembrane sheet. Scale bar is 100 nm. (B) Correspond-ing point-pattern analysis (red) and 99% confidence interval(black). c (cid:13)
Prior et al. (2003), originally published in
TheJournal of Cell Biology . doi:10.1083/jcb.200209091 [7].
Experimental evidence for the formation of nanoclus-ters (termed clusters from now on) is provided by in vivo and in vitro experiments. Fluorescence resonance energytransfer (FRET) studies show that activation by EGFleads to significant decrease in Ras lateral diffusion, sug-gesting the existence of Ras · GTP clusters [21]. A verysimilar result was obtained by single-molecule fluores-cence microscopy, where GTP-binding of Ras leads toslowly diffusing active Ras molecules [22]. Single particletracking (SPT) studies of fluorescently labeled Ras havealso demonstrated transient immobility of Ras (lastingless than 1s) with high temporal resolution, interspersedwith periods of free Brownian motion [23]. Furthermore,spatial statistics of fluorescently labeled Raf have shownthat Ras and Raf cluster together [24]. It is thereforebelieved that active Ras forms signaling platforms whichrecruit and activate Raf. As signaling platforms are Ras-isoform specific, the signal diversity observed between H-Ras, K-Ras and N-Ras is in part the result of differentialclustering properties in these isoforms [7].Direct evidence for protein clustering in a membranecan be obtained from high-resolution electron microscopy(EM). However, Ras is too small and not electron denseenough to be observed directly. To circumvent this prob-lem, Prior et al. used GFP-Ras fusion constructs whichwere treated with gold-labeled anti-GFP antibodies. Theresulting immunogold point patterns were visualized withEM (immunoEM) to quantitatively describe Ras cluster-ing (Fig. 1) [7]. It was found that the classical raftmodel, wherein a fixed number of lipid rafts accommo-date a fixed fraction of raft-inserted proteins, is incom-patible with the observed gold point patterns [11]. In-deed, for increasing expression levels, the classical modelpredicts an increase in the number of proteins per raft,and therefore a greater degree of clustering. To describethe data, Plowman et al. developed an alternative raftmodel, in which the size of Ras clusters remains constant.Assuming a constant, expression level-independent ratio
Ras density Ras per Gold density λ ras ( µm − ) lattice λ gold ( µm − )
625 225 2641,250 450 5252,500 900 1,0505,000 1,800 2,100
TABLE I: Representative Ras densities with correspondingnumbers of Ras molecules on discretized lattice membrane aswell as gold densities. Shown are the four Ras densities usedin Figs. 4, 5, and 6. For lattice parameters, see
Methods . of Ras in clusters to Ras monomers (cluster/monomerratio) results in the formation of more rafts as expres-sion increases, and supports the notion that lipidatedmolecules such as Ras can drive the formation of raftsin order to create signaling platforms [11]. This alter-native model predicts that 40% of active Ras moleculesform clusters of radius 6-12nm, each containing aboutseven Ras molecules, and 60% are randomly distributedmonomers [24]. While simulations of this model fit im-munoEM data, they do not provide a biophysical expla-nation for Ras clustering. Furthermore, these simulationsviolate laws of equilibrium physics. Specifically, the lawof mass action predicts an increase in the fraction of clus-tered molecules as the expression level is increased (untilmembrane saturates) [11]. This violation is troublesomeas the experiments are done on in vitro membrane sheets,where no active, energy-driven processes can limit clustersize. Membrane sheets were fixed (and proteins immobi-lized) after membrane removal from cells [7], leading toequilibration of membrane and proteins prior to imaging.Recent experiments even go further and probe the de-sign principles of signaling by Ras clusters. Such studiessuggest that, in the Ras-Raf-MEK-ERK pathway, Rasclusters act as an analog-digital-analog converter, whereanalog continuous EGF input is converted into digital,fully active clusters. The number of fully active Ras clus-ters, not the activity of individual Ras molecules, trans-lates into analog ERKpp output [25, 26]. Specifically,these experiments show that Ras mutants with wide-ranging activities lead to the same total cellular ERKppoutput [27]. This suggests that Ras clusters act as digi-tal nanoswitches, which become fully activated even forsmall inputs. Furthermore, the concentrations of activeRas and ERKpp are directly proportional to EGF input[24]. Hence, analog inputs produce analog outputs, me-diated by digital Ras clusters.Here we consider a physically-motivated model tostudy Ras clustering. The model mainly depends on aclose-contact, attractive interaction between active Rasmolecules (short range ∼ ∼ FIG. 2: Relation between L max and Ras density λ for im-munoEM data of gold labeled GFP-tH (black symbols) andRFP-tH (gray symbols), simulation averages and 99% confi-dence intervals (red), as well as a linear least-squares fit tosimulation averages (red line). L max data points were ex-tracted from Ref. [11] with IMAGE J . (Left inset) g max (black)and r max (green) as a function of λ . (Right inset) L max as afunction of λ without long-range repulsion ( V = 0). Errorbars represent standard deviations. For simulation details,including calculation of confidence intervals, see Methods . dent cluster/monomer ratio or the existence of lipid rafts,thus circumventing controversy surrounding their actual-ity. We equilibrate a discretized lattice membrane, occu-pied with active and inactive Ras molecules, using MonteCarlo simulations. After gold-labeling of Ras moleculesfrom simulation outputs, we perform a statistical cluster-ing analysis. The obtained statistical properties of Rasmolecules quantitatively agree with the statistical prop-erties of immuno-gold point patterns for wide-rangingRas expression levels (Fig. 2) [11]. Our model makespredictions about the signaling properties of Ras clus-ters, supporting the notion that Ras clusters indeed actas an analog-digital-analog converter [24]. II. RESULTS
Prior et al. [7] studied Ras clustering in plasmamembrane sheets using immunoEM of gold-labeled Rasmolecules (Fig. 1A). Gold point patterns were analyzedbased on Ripley’s K function. Specifically, the non-lineartransformation L ( r ) − r was applied where r is the dis-tance between gold particles. This function is zero forcomplete randomness, positive for clustering, and nega-tive for depletion (Fig. 1B). Plowman et al. [11] usedthe function’s maximal value, termed L max for short, assummary statistics for clustering and found that L max isindependent of Ras expression level (Fig. 2, symbols).This was rationalized by an ad hoc clustering model, as-suming a constant cluster/monomer ratio. Analysis ofimmuno-gold patterns is consistent with small clusters, FIG. 3: Model ingredients. (A) Short-range attraction (red)and long-range repulsion (blue) as a function of distance be-tween two Ras molecules for the parameters given in
Methods .Also shown is the cut-off beyond which the repulsive energyis set to zero (blue dashed line). (Inset) Representative partof lattice membrane showing three active Ras molecules (red)and one inactive Ras molecule (blue). Neighboring active Rasmolecules interact via the attractive short-range interaction(green bar). The cut-off used for the long-range repulsionis representatively shown for the central Ras (blue dashedcircle). (B) Schematic of a gold-labeled antibody associatedwith a GFP-Ras molecule in the inner leaflet of the plasmamembrane. containing approximately 6 to 8 molecules. Here we use abiophysical model of Ras clustering in the plasma mem-brane. In our model, a Ras molecule can be in eitheran active (on) or an inactive (off) state, correspondingto the respective GTP-bound and GDP-bound moleculesfor wild-type Ras. Both active and inactive Ras are as-sociated with membrane in line with experimental ob-servation [28]. The equilibrium probability of a singleRas molecule to be active depends on the effective free-energy difference between the on and off states, which inturn depends on input signals. We assume that activeRas molecules experience a short-range attraction, driv-ing cluster formation of active Ras, whereas a long-rangerepulsion limits cluster size (Fig. 3A, main panel). Sucha long-range interaction may result from lipid-anchor in-duced membrane deformations. To obtain equilibriumproperties, we approximate the membrane by a squarelattice, populated by Ras molecules of a specified den-sity (Fig. 3A, inset), and perform Monte Carlo simu-lations. For comparison with immunoEM experiments,we added 10nm-long gold-labeled antibodies (maximallyone per Ras) to the Ras molecules in the experimentallyobserved capture ratio (Fig. 3B). We mainly use the fourRas densities given in Table I. To specifically comparewith experiments on varying Ras-expression level (sym-bols in Fig. 2, main panel), we calculate the L max valuefor additional Ras densities. Note that these experimentsare based on the lipid anchor of H-Ras (tH), which hassimilar clustering properties as active H-Ras [11]. For de-tails on the experiments and our approach, see Methods .Figure 4 shows typical equilibrated membrane latticesfor the four Ras densities (left panels) with the corre-sponding plots of L ( r ) − r (right panels). For the low-est density, individual L ( r ) − r plots are highly variable.To produce meaningful statements about clustering we FIG. 4: Monte Carlo simulations and point-pattern analysis.Snapshots of equilibrated Ras molecules on lattice membrane(left column; active Ras in red and inactive Ras in blue) andcorresponding L ( r ) − r plots (right column) after gold labelingfor the four densities from Table I (density of Ras moleculesincreases from top to bottom). Shown in the L ( r ) − r plotsare individual simulations (cyan curves), their averages (thickblack curves), as well as 68.3%, 95.4%, and 99.0% confidenceintervals (red, green, and blue dashed lines, respectively). Forsimulation details, including calculation of confidence inter-vals, see Methods . also show the averaged plot, as well as provide confi-dence intervals. In line with experiment on varying Ras-expression level, we observe that for our model, L max isapproximately independent of Ras density (Fig. 2, mainpanel). The same is true if the analysis is done directly onRas molecules instead of the gold particles, demonstrat-ing the robustness of the result with respect to the detailsof Ras labeling by gold. Distance r max , defined as the dis-tance corresponding to L max , is about 8nm (Fig. 2, left FIG. 5: Monte Carlo simulations and point-pattern analysisfor conventional clustering model without long-range repul-sion ( V = 0). For a description of symbols and lines, see Fig.4. inset), or equivalently, 4 Ras molecules. Hence, clusterscontain few, about 4 to 10, Ras molecules. An alternativeclustering analysis based on the pair-correlation function g ( r ) gives similar results, i.e. g max values are indepen-dent of λ (Fig. 2, left inset). Hence, cluster sizes andtheir dependence on expression level are in good agree-ment with previous estimates [11].We also explored a more conventional clustering modelwithout the long-range repulsion, but maintaining theshort-range attraction. As shown in Fig. 5 (left pan-els), Ras molecules form increasingly larger clusters atincreasing Ras densities. Examination of the L ( r ) − r plots (right panels) shows that L max decreases for in-creasing Ras densities (Fig. 2, right inset), which is instark contrast to experiments. Hence, limiting the clustersize by the long-range repulsion is a necessary ingredientto correctly describe immunoEM data and, hence, Rasclustering.Next we examined the fraction of Ras molecules in clus-ters. Previous models assumed that the fraction is con-stant, i.e. independent of Ras density. In contrast, Fig.6 shows that in our model the distribution of the fractionclustered increases significantly with density, indicatingthat a constant cluster/monomer ratio is not required todescribe the immunoEM data in Fig. 2. Also shown inthe Fig. 6 is the fraction clustered for the conventionalclustering model without the long-range repulsion. Thedistribution also shifts to higher values with density, al-though to a lesser extent as fractions are much higherto start out with due to the missing long-range rangerepulsion. To clearly rule out the conventional cluster-ing model as a suitable model, we tested whether as-suming a constant fraction clustered (or equivalently, aconstant cluster/monomer ratio) can explain the immu-noEM data. For this purpose we collected simulationsfrom different densities but same fraction clustered andcompared their L max values. However, even with thisstrong selectivity of simulations, L max values continuedto decrease with increasing density (Fig. 6, inset).Figure 7 shows the signaling characteristics of Ras clus-ters for four different inputs. For input we use the free-energy difference between on (active) and off (inactive)Ras states ( cf. Eq. 1). To test if our model producesdigital-like nanoswitches, which are fully active even forsmall inputs, we identified clusters of two or more con-nected Ras molecules and calculated the cluster activ-ity, i.e. the fraction of active Ras molecules in clusters.The bar chart in Fig. 7A shows that our model indeedproduces nanoswitches, which are fully active even forsmall stimuli, as indicated by experiments [27]. In con-trast, the activity of a single Ras molecule does not be-have like a switch (Fig. 8A, dashed line). Furthermore,Fig. 7B provides the total activity of all Ras moleculesin the membrane irrespective of whether Ras moleculesbelong to clusters or not. We find the total activity isapproximately proportional to the input (black line) inthe range considered here in line with experiment [24].In our model, this is due to the fact that the number ofRas clusters is proportional to the input (Fig. 7B, blueline).There has recently been immense interest in under-standing the effect of noise in signal transduction [29, 30].Biochemical reactions are inherently noisy as they arebased on random collisions of molecules. This intrin-sic noise is further enhanced by the small number ofmolecules involved. Furthermore, rate constants mayfluctuate, as they depend on external conditions suchas other molecules not explicitly considered as part ofthe biochemical reactions. This extrinsic noise also in-cludes fluctuations in the input itself. To address howRas signaling is affected by noise, we compare signalingby Ras clusters (Fig. 7) with signaling by non-interactingRas molecules without clustering ability (Fig. 8). In-trinsic noise is inherently part of our simulations as Rasis allowed to randomly switch between the active and
FIG. 6: Distributions of Ras fractions in clusters. Differentcolors correspond to the four Ras densities from Table I, i.e. λ = 625 (red), λ = 1250 (blue), λ = 2500 (green), λ = 5000(black) in units of µ m − . Shown are results with (solid lines)and without (dashed lines) long-range repulsion. A Ras clus-ter is defined as two or more connected Ras molecules. (Inset) L max for pairwise constant fractions (overlapping fractions),i.e. fraction range 0.72-0.75 for λ = 625 and 1250 (circles anddashed line), fraction range 0.81-0.84 for λ = 1250 and 2500(triangles up and dotted line), and fraction range 0.88-0.91 for λ = 2500 and 5000 (triangles down and dashed-dotted line). the inactive states. The intrinsic noise for the activityof Ras in clusters (Fig. 7A, black error bars) is signifi-cantly less than for the activity of a single Ras molecule(Fig. 8A, black error bars) since clusters are fully activeand hence suppress random switching. This difference inintrinsic noise is reduced when considering the intrinsicnoise of the total activity from all Ras molecules in themembrane, which is only slightly smaller for Ras clus-ters (Fig. 7B, black error bars) than for non-interacting( V = J = 0) Ras molecules (Fig. 8B, black error bars).This is due to the fact that the number of Ras clus-ters, which is necessarily smaller than the number of Rasmolecules, can fluctuate significantly (Fig. 7B, blue errorbars). Most importantly, Ras clusters are more robust toinput noise, at least for sufficiently large inputs, thannon-interacting Ras molecules (by comparison of greenerror bars in Fig. 7A and Fig. 8A). Here, input noiserepresents fluctuations in input much faster than assem-bly/disassembly of clusters but slower than Ras signal-ing. Therefore, it is assumed that extrinsic noise onlyaffects the activity of Ras molecules, not clustering itself(see captions of Fig. 7A and Fig. 8A for details). Thisshows that Ras clusters have superior signaling proper-ties compared to non-interacting Ras molecules withoutclustering ability. FIG. 7: Signaling properties of Ras clusters. (A) Clusteractivity as a function of input (parameter ∆ ǫ ). Cluster activ-ity is defined as fraction of active Ras in clusters from sim-ulations (bar chart), where a cluster contains two or morecontacting Ras molecules. Also shown is approximate clus-ter activity P on = 1 / [1 + exp( N ∆ ǫ )], which assumes thatall N Ras molecules in a cluster (here chose N = 10) aretightly coupled and hence are either all on (active) or all off(inactive) together (black line). Black error bars show stan-dard deviation and represent intrinsic noise. Green error barsrepresent extrinsic noise, calculated with noise propagationformula δP on = ( dP on /d ∆ ǫ ) δ [∆ ǫ ] = NP on (1 − P on ) δ [∆ ǫ ] for δ [∆ ǫ ] = 0 . k B T . (B) Total activity of all Ras moleculesin the membrane, normalized by the total number of Rasmolecules (grey bar chart, left axis), and number of Ras clus-ters (white bar chart, right axis). Also shown are linear fits.Error bars represent standard deviations. To enlarge blackerror bars for better visualization in B , we used the square-root of the total variance from pooled simulations of inputs∆ ǫ , ∆ ǫ + 0 . k B T , and ∆ ǫ − . k B T . III. DISCUSSION
Different Ras isoforms are known to form nonoverlap-ping signaling clusters [8, 18], important for localizedsignaling of the Ras-Raf-MEK-ERK pathway [31, 32],involved in cell proliferation, differentiation, and apop-tosis [20]. In addition to the fundamental importance ofRas in this pathway, Ras mutations are found in 30% ofhuman cancers [13, 14, 15]. Ras clusters are also consid-ered evidence of lipid rafts [11]. Lipid rafts have attractedconsiderable interest due to their alleged role in proteinsorting and specificity of signaling [1, 2, 3]. In this work,we provided a biophysical model of Ras clustering, andcompared results with gold-point patterns obtained fromimmunoEM of plasma membrane extracts (Fig. 1). Inparticular, we obtained that clustering of Ras molecules,i.e. the cluster/monomer ratio, is independent of expres-sion level (Fig. 2), in line with experiments on the lipid
FIG. 8: Signaling properties of non-interacting Ras molecules.(A) Activity of single Ras molecule (dashed line; calculatedwith Eq. 1 for P on ) as a function of input (parameter ∆ ǫ ).Black error bars represent intrinsic noise, calculated from thesquare-root of the binomial variance P on (1 − P on ). Green er-ror bars are approximately 0.01 in magnitude and representextrinsic noise, calculated with the noise propagation formula δP on = P on (1 − P on ) δ [∆ ǫ ] and δ [∆ ǫ ] = 0 . k B T . (B) Totalactivity of all Ras molecules in the membrane, normalized bythe total number of Ras molecules (bar chart) and linear fit(dashed line). Error bars represent standard deviation, calcu-lated from the square-root of the total variance from pooledsimulations of inputs ∆ ǫ , ∆ ǫ + 0 . k B T , and ∆ ǫ − . k B T . anchor of H-Ras (tH) [11]. In our model, as well as in ex-periments, clustering is quantified by the maximum value(termed L max ) of function L ( r ) − r [11], where r is thedistance between gold particles. Our model has two mainingredients exemplified in Fig. 3: (1) a short-range at-traction between active Ras molecules (e.g. Ras · GTP)promoting clustering, and (2) a long-range repulsion be-tween Ras molecules, which limits cluster size.Another important feature, which makes our modelfundamentally different from previous Ras clusteringmodels [11, 24], is that the fraction of clustered Rasmolecules is not a model parameter [11] but a predictionfrom our simulations. Indeed, if we calculate the fractionof clustered molecules for the four densities from TableI, we obtain the distributions shown in Fig. 6. The frac-tion of clustered Ras increases with density, indicatingthat the assumption of a constant cluster/monomer ratio[11, 24] is misleading for describing the immunoEM datafor different expression levels [11]. Since this assump-tion violates equilibrium thermodynamics, our model ismore suitable for describing in vitro immunoEM data inabsence of energy sources from the cell. Note that inliving cells clustering may in part be regulated by ac-tive, energy-dependent mechanisms. For instance, clus-tering and signaling of constitutively active K-RasG12Vdepends on the presence of actin fences [11]. Suchmembrane-associated actin filaments, part of the actincortex, are highly dynamic and, hence, K-Ras cluster-ing may be regulated. An expression level-independentcluster/monomer ratio has also been found for glycosyl-phosphatidylinositol-anchored proteins (GPI-AP) in vivo [33]. Finally, clustering of proteins in the immunologicalsynapse is an active, actin-myosin dependent process [34],presumably to overcome the entropic barrier of localizingproteins [35].What is the role of Ras clusters beyond simple pro-tein sorting? Harding et al. argued that Ras clustersallow for highly precise coding of time-dependent in-puts, termed high fidelity signaling [25, 26]. First, Rasis highly abundant in the membrane (tens of thousandsmolecules), hence the number of active clusters can bewide-ranging depending on input, e.g., EGF. Second,clusters have a short life time (about 0 . · GTP) and ERKpp are proportional to EGF in-put [24]. Such analog ERK activation was recently alsoobserved in proliferating mammalian fibroblasts [36].The above listed properties of Ras clusters are sup-ported by our model. According to Fig. 7A, Ras clus-ters are fully active, even for small inputs. Nevertheless,the number of Ras clusters and hence the total activ-ity of all Ras molecules in the membrane are approx-imately proportional to the input (Fig. 7B), allowingfaithful transmission of continuous, time-dependent in-put signals. Interestingly, the activity of a single Rasmolecule and of non-interacting Ras molecules are alsoapproximately proportional to the input (Fig. 8A andFig. 8B, respectively). However, signaling by Ras clus-ters is less noisy and, hence, Ras clusters can transmitsignals more robustly than non-interacting Ras moleculeswithout clustering ability. The activity of Ras in clustersexhibits smaller intrinsic noise from random switchingbetween active and inactive states (Fig. 7A and B, blackerror bars). The activity of Ras in clusters is also lesssensitive to extrinsic noise from fast fluctuations in in-put than non-interacting Ras molecules (by comparisonof green error bars in Fig. 7A and Fig. 8A, respec-tively). The reason for the noise reduction by clustersis that Ras clusters are fully active, suppressing random switching between active and inactive states, as well asactivity changes due to fluctuations in input.Our model relies on the short-range attraction andlong-range repulsion between Ras molecules. The physi-cal origin of these interactions are yet to be determined.However, the attraction may originate from direct Ras-Ras interaction via hydrophobic, van der Waals, or elec-trostatic interactions [37], but may also be mediatedindirectly by scaffold proteins and lipids [17, 23, 37].The latter mechanism is supported by the finding thatthe positively-charged polybasic C-terminus of K-Rasbinds negatively charged phospholipids and sequestersacidic phospholipids, which may attract even more K-Rasmolecules. Furthermore, that mutant GFP-K-RasG12VS181D has a reduced ability to bind the membrane, aswell as to cluster [37]. Such a lipid-mediated mechanismwould support the concept of dynamic lipid rafts, whichonly form in presence of an activated membrane proteinsuch as Ras · GTP. The physical origin of the repulsion isharder to pinpoint, but may result from induced mem-brane curvature as a result of insertion of the farnesyl-polybasic anchor of K-Ras or the farnesyl-palmitate an-chor of H-Ras into the inner leaflet of the membrane [17].Lipid-anchor induced membrane deformations are sup-ported by molecular dynamics simulations [38], and maylead to long-range repulsion [39]. Furthermore, recentexperiments explicitly show that smallGTPases (Arf) in-duce membrane curvature [40].There are certain shortcomings of the immunoEMdata, rendering the cluster analysis of immuno-gold datadifficult. In our simulations, the addition of gold parti-cles to completely random distributions of Ras moleculesproduced the following. If more than one gold-labeled an-tibody is allowed to bind a Ras molecule, provided thereare no steric clashes between gold particles, the L ( r ) − r plot still predicts Ras clustering. Varying the antibodylength systematically resulted in distance r max being ap-proximately equal to the length of the antibody. This isdue to a small fraction of Ras molecules being associatedwith multiple gold particles: these gold particles will bewithin two antibody lengths of each other, and approxi-mately one antibody length from each other on average,resulting in a peak in the L ( r ) − r plot. This suggeststhat, unless it is verified that each Ras can only binda single antibody (i.e. the anti-GFP antibody can onlybind a single epitope on GFP fused to a Ras molecule),immunoEM data can overestimate clustering. In con-trast, two clustered Ras molecules in contact with eachother could each interact with separate antigen-bindingregions of the same antibody, since an antibody has twoantigen-binding regions. In this case the cluster of twoRas molecules is unobservable by immunoEM, leading toan understimation of clustering in the gold point pat-terns. These issues would have to be addressed if im-munoEM studies are to form the basis of an accuratequantification of Ras clustering.In conclusion, a comprehensive description of Ras clus-tering is an essential step in the understanding of Rassignaling properties, and of small, inner-membrane GT-Pases in general. For instance, we showed that clusteringleads to robustness to noise, especially input noise (Figs.7 and 8). While the model we have analyzed fits thedata (Figs. 2 and 7), several questions remain unan-swered. For one, lack of high resolution structural infor-mation about clustered Ras molecules prevents us from“seeing” clearly into the physicochemical basis of clus-tering (only partial crystal structures of Ras moleculesexist [41]). There is also the possibility of regulation ofclusters from within utilizing specific lipids and scaffoldproteins, which has scarcely been addressed in the litera-ture thus far [17], but would provide critical details to theconstruction of an accurate model for clustering. Whilewe have shown that immunoEM data can aid in the vi-sualization of Ras clusters, data regarding the dynamicsof clustering are found in the form of SPT [23], FRET[21], fluorescence recovery after photobleaching (FRAP)[42], and single-molecule fluorescence microscopy stud-ies [22, 43]. These data remain to be integrated intomore detailed spatio-temporal Monte Carlo simulations,so that the exchange of proteins between freely diffus-ing monomers in the membrane and immobile clusterscan be investigated [44]. Interestingly, our biophysicalmodel of Ras clustering shares the long-range repulsiondue to elasticity with recent models of lipid microphaseseparation [39] and chemoreceptor clustering [45, 46] inbacteria. Hence, similar biophysical principles may gov-ern the clustering of very different types of proteins inprokaryotic and eukaryotic membranes. The latter mayinclude EGF and Fc γ receptors, which are believed toassociate with rafts or to form small clusters [47, 48]. IV. METHODS
Experimental immunoEM data
Relevant experiments are described in [7, 11, 24, 27].Briefly, Ras clustering was examined on intact 2-Dsheets of apical plasma membrane, ripped off fromadherent baby hamster kidney cells directly onto EMgrids. Ras-fluorescent protein fusion constructs wereused, including the minimal plasma membrane tar-geting motif (lipid anchor) of H-Ras fused to GFP(GFP-tH) or RFP (RFP-tH), as well as constitutivelyactive H-RasG12V and K-RasG12V. These are taggedusing affinity-purified polyclonal anti-GFP antibodies,conjugated with 4nm gold particles, and visualized usingelectron microscopy (immunoEM). The resulting pointpatterns of gold particles were analysed for clustering(see below for details). Ras isoform clustering was foundto depend differentially on membrane-associated actin[11], lipid-raft constituent cholesterol [7], and scaffoldproteins galectin-1, galectin-3, and Sur-8 [49]. For arecent review see [18].
Biophysical model
A Ras molecule in the membrane can be in either oneof two states, active (on) with energy ǫ on or inactive(off) with energy ǫ off [50]. For wild-type Ras, the active(inactive) state corresponds to Ras · GTP (Ras · GDP).More generally, the two states correspond to two differ-ent protein conformations, making the two-state modelapplicable to activity mutants and lipid anchors as well.For any such two level system, the probability for asingle Ras molecule to be active is P on = 11 + e ∆ ǫ , (1)where ∆ ǫ = ǫ on − ǫ off is the free-energy difference betweenthe active and inactive states. While Eq. 1 is not explic-itly used for our simulations as it describes the activity ofRas in absense of interactions, it builds intuition aboutparameter ∆ ǫ . This parameter is effectively determinedby the input signal of the pathway, e.g. EGF, exceptfor the Ras activity mutants and lipid anchors, where itdesribes an energetic bias in conformational state.To describe clustering of active Ras molecules, as di-rectly observed for K-Ras and H-Ras using in vivo FRET[21], we introduce short-range attraction J between ac-tive Ras molecules, driving cluster formation. In orderto limit cluster size, we introduce long-range repulsion V ( r ), where r is the distance between two Ras molecules.For the repulsive interaction energy, we use a Gaussianfunction as previously applied for describing microphaseseparation of lipid mixtures [39] V ( r ) = V · exp (cid:18) − r σ (cid:19) , (2)where V is the maximal repulsion for two Ras moleculesin close proximity and σ is the width, i.e. the rangeof the repulsion beyond which the potential dropsquickly (Fig. 3A). The frustration between short-range attraction and long-range repulsion leads tosmall clusters. More precisely, the optimal clustersize corresponds to the minimum of the cluster energydivided by the number of Ras molecules in the cluster,i.e. the energy density [51]. The parameters usedin this study are ∆ ǫ = − . J = − . V = 2 . σ = 2nm. Long-range repulsions were neglectedbeyond a 6nm cut-off to reduce the calculation time. Allenergies are in thermal energy units k B T with k B beingthe Boltzmann constant and T the absolute temperature. Monte Carlo simulations
Since the immunoEM data are obtained from in vitro membrane sheets, clustering is an equilibrium process.Models of such phenomena are therefore particularlyamenable to Monte Carlo simulations, which include en-ergetics as well as entropy [51]. To set up simulations, wediscretize a defined area of the plasma membrane innerleaflet to obtain a two-dimensional M × M square-latticewhere M is the lattice size. On the lattice, each positionis uniquely described by an index i (if the 2-D lattice isthought of as a linear array of length M ). We assign aBoolean value s i to every Ras, where s i = 0 if Ras i isactive, and s i = 1 if Ras i is inactive. Using this nota-tion, we can construct an energy function describing thetotal energy E for a set of N molecules E = N X i =1 ∆ ǫ · (1 − s i ) + X h i,j i J · (1 − s i )(1 − s j ) + X i,j V ( r ij )(3)where h i, j i denotes nearest-neighbor pairs.After randomly generating the positions of the startingRas molecules on the lattice, individual Ras moleculesare chosen at random and attempted to move to anew location on the lattice. Included in each step isa probability of switching between active and inactiveRas. Moves are accepted or rejected based on theMetropolis-Hastings algorithm. We use a lattice of size M = 300 and a lattice constant a = 2nm (the size of aRas molecule [44]), resulting in a 0 . µ m membrane.In order to reduce boundary effects of the lattice, weadopt periodic boundary conditions. Gold particles
In order to compare the simulation outputs with immu-noEM data, gold-labeled antibodies are added to theequilibrated Ras molecules. The length of the antibodyused in the experiments is 10nm [11]. When an antibodybinds a Ras molecule, the gold particle associated withthe antibody can at any one time occupy any positionon the surface of a hemisphere around the Ras molecule(Fig. 3B). For simplicity, the radius of the hemisphereis chosen equal to the length of the antibody, andthe centre of the gold particle’s position is projectedonto the plane of the membrane, defining the particle’sposition on the lattice. This position is then matchedagainst previous gold positions for steric clashes, andif it is found to be closer than 4nm from another goldparticle, the position is rejected and another Ras ispicked at random. This process is iterated until 42%of Ras molecules are occupied, corresponding to theexperimentally-observed capture ratio [11].
Cluster analysis
We use three functions to evaluate the degree of Ras clus-tering: K ( r ), L ( r ) − r and g ( r ). Ripley’s K -function K ( r )was first proposed for analyzing spatial point patterns[52]. K ( r ) calculates the expected number of particleswithin a distance r of any particle, normalized by the average density λK ( r ) = N ( r ) λ (4)= 1 λ N N X i =1 N X i = j I ij ( || x j − x i || ≤ r ) (5)= AN ( N − N X i =1 N X i = j I ij ( || x j − x i || ≤ r ) (6)where A is the area of the lattice studied, N the numberof Ras molecules or gold particles, λ the surface densityand I ij ( x ) an indicator function which takes a value of 1 if || x j − x i || ≤ r and 0 otherwise. Under the null hypothesisof complete spatial randomness, N ( r ) = λπr , so K ( r ) = πr . An often used non-linear transformation of K ( r )which we shall employ is [7] L ( r ) − r = r K ( r ) π − r (7)which has a value of 0 for complete spatial randomness,is positive for clustering, and negative for depletion ofparticles. For large r , L ( r ) − r is zero on average, sinceparticles are uncorrelated. Since L ( r ) − r is a non-lineartransformation of K ( r ), when averaging over multiplesimulations to resemble a large piece of membrane (seebelow), the K ( r ) values are averaged first and only thentransformed into L ( r ) − r . For further analyses of simu-lations, we use summary statistics L max = max[ L ( r ) − r ]and corresponding distance r max = argmax r [ L ( r ) − r ].The pair-correlation function g ( r ) can be defined intwo different ways. The first by normalizing and differ-entiating K ( r ) [53, 54] g ( r ) = 12 πr dK ( r ) dr (8)and the second by counting in a similar manner to K ( r )but in concentric rings: g ( r ) = AN ( N −
1) 12 πra N X i =1 N X i = j I ij ( r − a < || x j − x i || ≤ r (cid:1) (9)for r ≥ a , where a is the lattice constant. Testingthese two versions of the pair-correlation function yieldedslightly different absolute values of g ( r ), but the relativebehaviors of the two were identical. For a random distri-bution of particles and for large r in general, g ( r ) takes avalue of 1 on average. We again use summary statistics g max = max g ( r ) and r max = argmax r g ( r ).To estimate confidence intervals for the L ( r ) − r cluster analysis, 99 simulations were run for each densitywith all interactions set to zero, simulating a randomdistribution of Ras in order to obtain an estimate of thebackground clustering noise intrinsic to each density.Triplets of K ( r ) were averaged to simulate a 1 µ m L ( r ) − r valueswere calculated for each. The 68.3%, 95.4%, and 99.0%confidence intervals for individual measurements wereobtained by multiplying the standard deviation of the33 triplets by 1, 2, and 2.576, respectively. Standarddeviations for L max , g max , and r max were calculatedbased on triplets as well. Supporting Information
Accession Numbers
Acknowledgments
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